Question

1. in $\triangle abc$, the angle bisector of $\angle bac$ intersects $bc$ at $d$. which relationship does the angle bisector theorem establish? options: (a) formula, (b) $bd = dc$, (c) formula, (d) $ab = ac$ 5. a rectangle $r$ has vertices $p(0,0)$, $q(4,0)$, $r(4,2)$, and $s(0,2)$. it is transformed into $p(1,1)$, $q(3,1)$, $r(3,3)$, and $s(1,3)$. is the resulting figure similar, and what transformation was performed? options: (a) yes, the figures are similar; a translation was performed. (b) yes, the figures are similar; a rotation was performed. (c) no, the figures are not similar; a dilation was performed. (d) no, the figures are not similar; a reflection was performed. 6. which similarity criterion is used to prove that $\triangle xyw \sim \triangle xyz \sim \triangle ywz$ when $yw$ is the altitude to the hypotenuse? options: (a) aa, (b) sss, (c) sas, (d) hl.

Explanation:

Question 4

The Angle Bisector Theorem states that an angle bisector in a triangle divides the opposite side into segments proportional to the adjacent sides. In $\triangle ABC$, with angle bisector $AD$ of $\angle BAC$ intersecting $BC$ at $D$, the theorem gives $\frac{BD}{DC}=\frac{AB}{AC}$. Let's analyze the options:

• Option a: $\frac{BD}{DC}=\frac{AC}{AB}$ is incorrect as per the theorem.
• Option b: $BD = DC$ would be true for a median, not an angle bisector (unless the triangle is isoceles with $AB = AC$), so incorrect.
• Option c: $\frac{BD}{DC}=\frac{AB}{AC}$ matches the Angle Bisector Theorem, so correct.
• Option d: $AB = AC$ is not established by the Angle Bisector Theorem (it's true for isoceles triangles, but the theorem is about proportional segments, not equal sides), so incorrect.

First, analyze the transformation:

• Original rectangle vertices: $P(0,0)$, $Q(4,0)$, $R(4,2)$, $S(0,2)$. Length $PQ = 4$, width $PS = 2$.
• Transformed vertices: $P'(1,1)$, $Q'(3,1)$, $R'(3,3)$, $S'(1,3)$. Length $P'Q' = 3 - 1 = 2$, width $P'S' = 3 - 1 = 2$. Wait, no—wait, original rectangle has length 4 (horizontal) and width 2 (vertical). Transformed figure: distance between $P'(1,1)$ and $Q'(3,1)$ is $2$, between $Q'(3,1)$ and $R'(3,3)$ is $2$, so it's a square? Wait, no, original is rectangle (length 4, width 2), transformed: let's check the sides. Wait, maybe I miscalculated. Wait, original $PQ$: from (0,0) to (4,0): length 4. $QR$: (4,0) to (4,2): length 2. Transformed $P'Q'$: (1,1) to (3,1): length 2. $Q'R'$: (3,1) to (3,3): length 2. So original is rectangle (length 4, width 2), transformed is square (length 2, width 2). Wait, but the options mention similarity. Wait, no—wait, maybe it's a dilation? Wait, no, the coordinates: original $P(0,0)$ to $P'(1,1)$: translation? Wait, no, translation would be adding a vector. Wait, original $P(0,0)$ to $P'(1,1)$: vector (1,1). $Q(4,0)$ to $Q'(3,1)$: vector (-1,1). Wait, that's not translation. Wait, maybe I made a mistake. Wait, original rectangle: length 4 (x from 0 to 4), width 2 (y from 0 to 2). Transformed: x from 1 to 3 (length 2), y from 1 to 3 (width 2). So the original has length 4, width 2 (ratio 2:1), transformed has length 2, width 2 (ratio 1:1). So they are not similar? But the options: wait, no—wait, maybe it's a dilation with scale factor 0.5 and translation? Wait, no, the options are translation, rotation, dilation, reflection. Wait, the original is a rectangle, transformed: let's check angles. All angles are 90 degrees, so angles are equal. Now, check side ratios. Original length: 4, width: 2 (ratio 2:1). Transformed length: 2, width: 2 (ratio 1:1). So ratios are not equal, so not similar. But which transformation? Wait, the coordinates: $P(0,0)\to P'(1,1)$, $Q(4,0)\to Q'(3,1)$, $R(4,2)\to R'(3,3)$, $S(0,2)\to S'(1,3)$. Let's see the differences: $P$ to $P'$: (1,1); $Q$ to $Q'$: (-1,1); $R$ to $R'$: (-1,1); $S$ to $S'$: (1,1). Wait, that's a reflection? No, reflection over y=x? No. Wait, maybe it's a dilation with center (2,1) or something? Wait, the options: option c says "No, the figures are not similar; a dilation was performed." Wait, but if we dilate the original rectangle (length 4, width 2) by scale factor 0.5, we get length 2, width 1, but here the transformed width is 2. So maybe my analysis is wrong. Wait, original rectangle: $P(0,0)$, $Q(4,0)$, $R(4,2)$, $S(0,2)$. Transformed: $P'(1,1)$, $Q'(3,1)$, $R'(3,3)$, $S'(1,3)$. Let's calculate the side lengths:
• Original $PQ$: $4 - 0 = 4$; $QR$: $2 - 0 = 2$.
• Transformed $P'Q'$: $3 - 1 = 2$; $Q'R'$: $3 - 1 = 2$.

So original is rectangle (length 4, width 2), transformed is square (length 2, width 2). So the side ratios are not equal (original 4:2 = 2:1, transformed 2:2 = 1:1), so not similar. Now, the transformation: if we dilate the original rectangle by scale factor 0.5 around the center (2,1), let's see: center (2,1). Distance from center to $P(0,0)$: $\sqrt{(2 - 0)^2 + (1 - 0)^2} = \sqrt{5}$. Distance from center to $P'(1,1)$: $\sqrt{(2 - 1)^2 + (1 - 1)^2} = 1$. Wait, not 0.5. Alternatively, maybe it's a translation and dilation? But the options: option c says dilation. Wait, maybe the original problem has a typo, but among the options, if the figures are not similar and dilation was performed, option c. Wait, but let's check the options again:

• Option a: translation: translation preserves shape and size, so…

When $YW$ is the altitude to the hypotenuse of right triangle $XYZ$ (assuming $XYZ$ is right-angled, say at $Y$), then $\triangle XYW$, $\triangle XYZ$, and $\triangle YWZ$ are similar. The AA (Angle-Angle) similarity criterion is used:

• $\angle X$ is common to $\triangle XYW$ and $\triangle XYZ$, and $\angle XWY = \angle XYZ = 90^\circ$, so AA similarity.
• $\angle Z$ is common to $\triangle YWZ$ and $\triangle XYZ$, and $\angle ZWY = \angle ZYX = 90^\circ$, so AA similarity.
• Also, $\angle XYW = \angle Z$ (since in right triangle, acute angles are complementary), so $\triangle XYW \sim \triangle YWZ$ by AA.

So the similarity criterion is AA.

Answer:

c. $\boldsymbol{\frac{BD}{DC}=\frac{AB}{AC}}$

Question 5