Question

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circle with center y, radius yz = 21. line yb intersects the circle at a, with ab = 8. line zb is a secant or tangent? (assuming zb is tangent at z, so yz is radius, yz ⊥ zb. then we can use the tangent-secant theorem: zb² = ab (ab + 2ya) but ya = yz = 21, so zb = x + 8? wait, actually, if zb is tangent at z, then yz ⊥ zb, triangle yzb is right-angled at z. yz = 21, yb = ya + ab = 21 + 8 = 29? wait, no: ya is radius, so ya = 21, ab = 8, so yb = ya + ab = 21 + 8 = 29? then in right triangle yzb, yz = 21, yb = 29, so zb = √(yb² - yz²) = √(29² - 21²) = √((29 - 21)(29 + 21)) = √(850) = √400 = 20. but zb is x + 8? wait, no, z to b is x, and z to a? wait, maybe the diagram is: z is tangent point, b is outside, line bz is tangent at z, line ba is secant, intersecting circle at a and... wait, the diagram shows y as center, z on circle, a on circle, b outside. so zb is tangent at z, so yz ⊥ zb. yz = 21 (radius), yb = ya + ab = 21 + 8 = 29? wait, ya is radius, so ya = 21, ab = 8, so yb = 21 + 8 = 29. then in right triangle yzb, legs yz = 21, zb = x, hypotenuse yb = 29. so x² + 21² = 29². so x² = 29² - 21² = (29 - 21)(29 + 21) = 850 = 400, so x = 20. so the problem is to find x, the length from z to b (tangent segment) given that ab = 8, radius yz = 21. so using pythagoras in right triangle yzb, where yz is radius (21), yb is (21 + 8) = 29? wait, no, yb is the distance from center y to point b, which is ya + ab, but ya is radius (21), ab is 8, so yb = 21 + 8 = 29. then zb is the tangent segment, so yz ⊥ zb, so triangle yzb is right-angled at z. therefore, zb² + yz² = yb². so zb² = 29² - 21² = 841 - 441 = 400, so zb = 20. but in the diagram, z to b is x, and a is between y and b, so zb is x, and ab is 8, so maybe za is... wait, maybe the problem is to find x, where zb = x, and ab = 8, so the tangent-secant formula: zb² = ab (ab + 2radius)? no, tangent-secant theorem is: if a tangent from b touches the circle at z, and a secant from b passes through a and another point (but here the secant is ba, but a is on the circle, so the secant is ba, but ba is from b to a, with a on circle, so the secant length is ba, but thats only one intersection? wait, no, the secant should intersect the circle at two points. so maybe the diagram is: b is outside, line bz is tangent at z, line ba is secant, intersecting circle at a and another point, but in the diagram, a is between y and b, so yb is from center y to b, passing through a (on circle), so ya is radius (21), ab is 8, so yb = 21 + 8 = 29. then zb is tangent, so yz ⊥ zb, so triangle yzb is right-angled at z. therefore, zb = √(yb² - yz²) = √(29² - 21²) = 20. so x = 20. the ocr text is about the geometric diagram with circle, center y, radius 21, tangent zb, secant yb with ab = 8, find x (length zb).

Explanation:

Step1: Identify the tangent and secant

$BZ$ is a tangent to the circle at $Z$, and $BY$ is a secant intersecting the circle at $A$. So we use the tangent - secant rule: $BZ^{2}=BA\times BY$.
First, let $BA = 8$, $YA=21$, so $BY=BA + AY=8 + 21=29$? Wait, no, wait. Wait, $YZ$ is a radius, so $YZ = Y A=21$ (since both are radii). Wait, the tangent - secant formula is $BZ^{2}=BA\times BB'$? Wait, no, the correct formula is: If a tangent from $B$ touches the circle at $Z$, and a secant from $B$ passes through $A$ (where $A$ is on the circle) and $Y$ (the center? No, $Y$ is the center, so $YA$ and $YZ$ are radii, length 21. So the secant segment is $BA$ and $BY$? Wait, no, the secant segment is from $B$ to $A$ (external part) and from $B$ to the second intersection point. Wait, actually, the tangent - secant theorem states that if a tangent from an external point $B$ touches the circle at $Z$, and a secant from $B$ passes through the circle, intersecting it at $A$ and $C$ (but in this case, the secant is $BY$, with $A$ between $B$ and $Y$? Wait, no, $Y$ is the center, so $YA$ is a radius, so $YA = 21$, and $BA = 8$, so the length of the secant from $B$ to the circle is $BA$ (external part) and $BY=BA + AY=8 + 21=29$? Wait, no, the tangent - secant formula is $BZ^{2}=BA\times BB'$, where $BB'$ is the entire secant length. Wait, actually, the formula is: If a tangent from $B$ touches the circle at $Z$, and a secant from $B$ intersects the circle at $A$ and $Y$ (with $A$ between $B$ and $Y$), then $BZ^{2}=BA\times BY$. Wait, $YZ$ is a radius, so $YZ = 21$, and $BZ$ is tangent, so $\triangle YZB$ is right - angled at $Z$ (tangent is perpendicular to radius). Wait, maybe I made a mistake earlier. Let's correct:

Since $YZ$ is a radius and $BZ$ is a tangent, $\angle YZB = 90^{\circ}$. So by Pythagoras theorem in $\triangle YZB$, $BZ^{2}+YZ^{2}=BY^{2}$. But also, by the tangent - secant theorem, $BZ^{2}=BA\times BY'$, where $BY'$ is the secant. Wait, no, the tangent - secant theorem is: If a tangent from $B$ touches the circle at $Z$, and a secant from $B$ passes through the circle, intersecting it at $A$ (first intersection) and $Y$ (second intersection, where $Y$ is outside? No, in the diagram, $Y$ is the center, so $A$ is between $B$ and $Y$. So the external segment is $BA = 8$, and the entire secant segment is $BY=BA + AY=8 + 21=29$? Wait, no, $AY$ is a radius, so $AY = 21$, so $BY=BA + AY=8 + 21=29$. Then by tangent - secant theorem, $BZ^{2}=BA\times BY$. But $BZ$ is also the tangent, and $YZ$ is radius, so in right triangle $YZB$, $BZ^{2}=BY^{2}-YZ^{2}$. Wait, this is a contradiction unless my understanding is wrong. Wait, no, the tangent - secant theorem is: If a tangent from $B$ touches the circle at $Z$, and a secant from $B$ intersects the circle at $A$ and $C$ (with $A$ closer to $B$), then $BZ^{2}=BA\times BC$. In this case, the secant is $BA$ (external part) and $BC = BA + AC$. But here, the center is $Y$, so $AC$ would be the diameter? No, $YA$ and $YZ$ are radii, so $YA=21$, so the secant from $B$ passes through $A$ (on the circle) and $Y$ (the center, which is inside the circle? Wait, no, the center is inside the circle, so the secant from $B$ (outside the circle) passes through $A$ (on the circle) and then goes to $Y$ (inside the circle). Wait, that can't be. So my initial approach was wrong.

Wait, let's re - examine the diagram. $Y$ is the center, so $YA$ and $YZ$ are radii, length 21. $B$ is outside the circle, $BZ$ is tangent at $Z$, $BA$ is a segment from $B$ to $A$ (where $A$ is on the circle), and $BY$ is a segment from $B$ to $Y$ (the center). So the length of the secant from $B$ to the circle is $BA$ (external part) and the length from $A$ to the other intersection point. But since $Y$ is the center, the other intersection point would be on the line $BY$ extended beyond $Y$, but in the diagram, $A$ is between $B$ and $Y$. So actually, the correct formula is the power of a point. The power of point $B$ with respect to the circle is equal to the square of the length of the tangent from $B$ to the circle, and also equal to $BA\times BB'$, where $BB'$ is the length of the secant from $B$ through the circle. Wait, the power of a point $B$ outside the circle is $BZ^{2}=BA\times BY'$, where $BY'$ is the length of the secant from $B$ to the second intersection point $Y'$ (on the circle, on the other side of $Y$ from $A$). But in this case, since $Y$ is the center, the distance from $A$ to $Y$ is 21, so the distance from $B$ to $Y$ is $BA + AY=8 + 21=29$, and the distance from $B$ to the other intersection point (let's say $C$) would be $BY + YC=29 + 21=50$? No, that doesn't make sense.

Wait, maybe the diagram is such that $BY$ is a secant where $A$ is a point on $BY$ and on the circle, and $Z$ is the point of tangency. So the power of point $B$: $BZ^{2}=BA\times BB'$, where $BB'$ is the length of the secant from $B$ to the circle, passing through $A$ and then to the other side. Wait, no, the correct formula for the power of a point $B$ outside the circle is: If a tangent from $B$ touches the circle at $Z$, and a secant from $B$ intersects the circle at $A$ and $C$ (with $A$ between $B$ and $C$), then $BZ^{2}=BA\times BC$. In this case, $BC$ is the length from $B$ to $C$ (the second intersection point), and $BA$ is from $B$ to $A$. So if $A$ is between $B$ and $Y$, and $Y$ is the center, then $C$ would be on the line $BY$ extended beyond $Y$, so $BC=BA + AY+YC$. But $AY = YC = 21$ (radii), so $BC=8 + 21+21 = 50$, and $BA = 8$, then $BZ^{2}=8\times50 = 400$, so $BZ = 20$, but that doesn't match with the Pythagorean approach. Wait, no, I think I messed up the power of a point.

Wait, let's start over. The power of point $B$ with respect to the circle with center $Y$ is defined as $BZ^{2}$ (since $BZ$ is tangent) and also as $BY^{2}-r^{2}$, where $r$ is the radius (because in the right triangle $YZB$, $BZ^{2}+r^{2}=BY^{2}$, so $BZ^{2}=BY^{2}-r^{2}$). Also, by the power of a point theorem, if a secant from $B$ passes through $A$ (on the circle), then the power of $B$ is also $BA\times BB'$, where $BB'$ is the length of the secant from $B$ to the second intersection point. But in this case, the secant is $BA$ (external segment) and $AA'$ (the chord), but since $Y$ is the center, the line $BY$ passes through $A$ and $Y$, so the length of the secant from $B$ to the circle is $BA$ (from $B$ to $A$) and then from $A$ to the other side of the circle through $Y$, which is $A$ to $C$ where $AC = 2r=42$ (since $A$ is on the circle and $Y$ is the center, $AY = 21$, so $AC = 2\times21 = 42$). Wait, no, $A$ is on the circle, so the distance from $A$ to $Y$ is 21 (radius), so the length of the secant from $B$ is $BA + AY+YC$? No, $A$ is on the circle, so the secant is from $B$ to $A$ (length $BA = 8$) and then from $A$ to the other intersection point, which is $A$ to $C$ where $C$ is on the line $BY$ and $YC = 21$ (since $Y$ is the center). Wait, no, the correct power of a point formula for a point outside the circle is: If a tangent from $B$ touches the circle at $Z$, and a secant from $B$ intersects the circle at $A$ and $C$ (with $A$ closer to $B$), then $BZ^{2}=BA\times BC$. Here, $BC=BA + AC$, and $AC$ is the length of the chord through $A$ and $C$. Since $Y$ is the center, $AC$ is a chord, and $AY$ is a radius, so if $A$ is on the circle, then $AC$ can be any chord, but in this case, the secant is along $BY$, so $AC = 2\times AY=42$ (diameter). Wait, no, $AY$ is 21 (radius), so the distance from $A$ to $Y$ is 21, so the distance from $A$ to the other end of the diameter through $A$ is $2\times21 = 42$. So $BC=BA + AC=8 + 42=50$. Then $BZ^{2}=BA\times BC=8\times50 = 400$, so $BZ = 20$. But also, in $\triangle YZB$, $YZ = 21$ (radius), $BY=BA + AY=8 + 21=29$, and by Pythagoras, $BZ^{2}+YZ^{2}=BY^{2}\implies BZ^{2}+21^{2}=29^{2}\implies BZ^{2}=29^{2}-21^{2}=(29 - 21)(29 + 21)=8\times50 = 400$, which matches. So then, in the diagram, $BZ=x$, $YZ = 21$, $BY=29$, so by Pythagoras, $x^{2}+21^{2}=29^{2}\implies x^{2}=29^{2}-21^{2}=841 - 441 = 400\implies x = 20$. Wait, but the problem is to find $x$, which is $BZ$? Wait, no, the diagram shows $Z$ to $B$ as $x$, $B$ to $A$ as 8, $Y$ to $A$ as 21, $Y$ to $Z$ as 21.

Wait, maybe the question is to find $x$, the length of $ZB$. So:

Step1: Apply Pythagoras in $\triangle YZB$

Since $BZ$ is tangent to the circle at $Z$, $\angle YZB = 90^{\circ}$. So by Pythagoras theorem:
$BZ^{2}+YZ^{2}=BY^{2}$

Step2: Determine the lengths of $YZ$ and $BY$

$YZ$ is a radius, so $YZ = 21$.
$BY=BA + AY=8 + 21 = 29$ (since $BA = 8$ and $AY = 21$)

Step3: Solve for $BZ$ (which is $x$)

Substitute the values into the Pythagorean formula:
$x^{2}+21^{2}=29^{2}$
$x^{2}=29^{2}-21^{2}$
We know that $a^{2}-b^{2}=(a - b)(a + b)$, so:
$x^{2}=(29 - 21)(29 + 21)$
$x^{2}=8\times50$
$x^{2}=400$
Take the square root of both sides:
$x=\sqrt{400}=20$

Answer:

20