Question

triangle abc and triangle qrs are similar and are the same orientation. the endpoints of the hypotenuse of triangle abc are a(-6, 8) and b(-2, 2). the hypotenuse of triangle qrs is on the same line as the hypotenuse of triangle abc and is one - half the length of ab.
what is the slope of the hypotenuse of triangle abc?
what are the coordinates of the hypotenuse of triangle qrs?
q(2, -4) and r( , )

Explanation:

Step1: Calculate slope of AB

The slope formula is $m = \frac{y_2 - y_1}{x_2 - x_1}$. For points $A(-6, 8)$ and $B(-2, 2)$, substitute into the formula: $m = \frac{2 - 8}{-2 - (-6)} = \frac{-6}{4} = -\frac{3}{2}$.

Step2: Analyze QRS hypotenuse

Since QRS hypotenuse is on the same line as AB (same slope) and half its length. The vector from A to B is $(-2 - (-6), 2 - 8) = (4, -6)$. Half of this vector is $(2, -3)$. Starting from Q(2, -4), add this vector: $x = 2 + 2 = 4$? Wait, no—wait, the direction: AB goes from A(-6,8) to B(-2,2). The midpoint? Wait, no, the length of AB: distance is $\sqrt{(4)^2 + (-6)^2} = \sqrt{16 + 36} = \sqrt{52}$. Half length would be half the vector. Wait, Q is (2, -4). Let's find R. The slope of AB is $-\frac{3}{2}$, so the line for QRS hypotenuse has slope $-\frac{3}{2}$. The vector from A to B is (4, -6), so half vector is (2, -3). But since QRS is same orientation, we can also think of moving from Q in the direction of AB's vector scaled by 1/2. Wait, AB's vector is (4, -6), so half is (2, -3). So from Q(2, -4), adding (2, -3) gives (4, -7)? No, that doesn't seem right. Wait, maybe the midpoint? Wait, no, the problem says "the hypotenuse of triangle QRS is on the same line as the hypotenuse of triangle ABC and is one - half the length of AB". So first, slope of AB is $-\frac{3}{2}$. For the coordinates of R: since Q is (2, -4), and the vector from A to B is (4, -6), the vector for QR should be half of AB's vector, but in the same direction. Wait, AB's vector is ( - 2 - (-6), 2 - 8)=(4, -6). Half vector is (2, -3). So starting at Q(2, -4), add (2, -3): x = 2 + 2 = 4, y = -4 + (-3)= -7? No, that can't be. Wait, maybe the other way: AB goes from A(-6,8) to B(-2,2). The line equation: $y - 8 = -\frac{3}{2}(x + 6)$. Simplify: $y = -\frac{3}{2}x - 9 + 8 = -\frac{3}{2}x - 1$. Now, Q is (2, -4). Let's check if Q is on this line: $-\frac{3}{2}(2) - 1 = -3 -1 = -4$. Yes! So Q is on the line. Now, the length of AB: distance between (-6,8) and (-2,2) is $\sqrt{(4)^2 + (-6)^2}=\sqrt{16 + 36}=\sqrt{52}=2\sqrt{13}$. Half length is $\sqrt{13}$. So the distance from Q to R should be $\sqrt{13}$. Let R = (x,y). Then $\sqrt{(x - 2)^2 + (y + 4)^2}=\sqrt{13}$, and $\frac{y + 4}{x - 2}=-\frac{3}{2}$. From the slope: $y + 4 = -\frac{3}{2}(x - 2)$. Substitute into distance: $(x - 2)^2 + (-\frac{3}{2}(x - 2))^2 = 13$. Let $u = x - 2$, then $u^2 + \frac{9}{4}u^2 = 13 \Rightarrow \frac{13}{4}u^2 = 13 \Rightarrow u^2 = 4 \Rightarrow u = 2$ or $u = -2$. Since the triangles are same orientation, and AB goes from A(-6,8) to B(-2,2) (right and down), QRS should go from Q(2, -4) in the same direction (right and down), so u = 2. Thus, x - 2 = 2 $\Rightarrow$ x = 4, y + 4 = -\frac{3}{2}(2) $\Rightarrow$ y + 4 = -3 $\Rightarrow$ y = -7? Wait, no, that's down. Wait, but maybe u = -2? Then x - 2 = -2 $\Rightarrow$ x = 0, y + 4 = -\frac{3}{2}(-2)= 3 $\Rightarrow$ y = -1. Wait, let's check distance: (0 - 2)^2 + (-1 + 4)^2 = 4 + 9 = 13, which is correct. And slope: (-1 + 4)/(0 - 2)= 3/(-2)= -3/2, correct. Now, which direction is same orientation? Triangle ABC: A is top, B is bottom - right of A. Triangle QRS: Q is (2, -4), so same orientation would mean R is bottom - right of Q? Wait, A(-6,8) to B(-2,2): x increases by 4, y decreases by 6. Q(2, -4): x increases by 2 (half of 4), y decreases by 3 (half of 6) would be (4, -7), but that distance is $\sqrt{(2)^2 + (-3)^2}=\sqrt{13}$, correct. Wait, earlier when u = 2, x = 4, y = -7: distance from Q(2, -4) is $\sqrt{(2)^2 + (-3)^2}=\sqrt{13}$, correct. And slope is (-7 + 4)/(4 - 2)= (-3)/2 = -3/2, correct. But when u = -2, x = 0, y = -1: x decreases by 2, y increases by 3, which is opposite direction. So same orientation means same direction as AB (x increases, y decreases), so R should be (4, -7)? Wait, but let's check the graph. Wait, the graph of ABC: A is at (-6,8), B at (-2,2), C at (-6,2)? Wait, no, the graph shows C at (-6,2) and B at (-2,2), A at (-6,8). So AB is from (-6,8) to (-2,2), so right 4, down 6. Q is (2, -4), so right 2, down 3 would be (4, -7), which is same direction. So R is (4, -7)? Wait, but earlier when u = -2, it's left 2, up 3, which is opposite. So same orientation: R should be (4, -7). But wait, let's re - evaluate the vector. AB vector: ( - 2 - (-6), 2 - 8)=(4, -6). Half vector: (2, -3). So Q(2, -4) + (2, -3)=(4, -7). Yes, that's correct.

Answer:

Slope of hypotenuse of triangle ABC: $\boldsymbol{-\frac{3}{2}}$
Coordinates of R: $\boldsymbol{(4, -7)}$ (Wait, but earlier when u = -2, we got (0, -1), but same orientation should be same direction. Wait, let's check the problem statement: "the same orientation". So triangle ABC: A is top, C is bottom - left, B is bottom - right. So triangle QRS should have Q as top? Wait, no, Q is (2, -4), so maybe Q is top, R is bottom - right? Wait, A(-6,8) (top), B(-2,2) (bottom - right), C(-6,2) (bottom - left). So triangle QRS: Q(2, -4) (top), R (bottom - right), S (bottom - left). So the hypotenuse is QR, same as AB. So vector from Q to R should be same direction as A to B: (4, -6) scaled by 1/2: (2, -3). So Q(2, -4) + (2, -3)=(4, -7). So R is (4, -7).