Question

1. a diagram of similar triangles is shown below. 43 x 32 16 what is the value of x, in feet?

Explanation:

Step1: Identify similar triangles

The two right triangles are similar (by AA similarity, as both are right - angled and the vertical angles are equal). For similar triangles, the ratios of corresponding sides are equal. So we have the proportion $\frac{43}{16}=\frac{x + 32}{32}$? Wait, no, let's re - examine the diagram. Wait, actually, the sides of the first right triangle are 43 (horizontal), $x$ (vertical), and hypotenuse. The second right triangle has horizontal side 16 and vertical side 32, and hypotenuse. Since the triangles are similar, the ratio of horizontal side to vertical side of the first triangle should be equal to the ratio of horizontal side to vertical side of the second triangle? Wait, no, let's set up the proportion correctly. Let's assume that the two triangles are similar, so $\frac{43}{x}=\frac{16}{32}$? No, that's not right. Wait, maybe the correct proportion is $\frac{43}{16}=\frac{x}{32}$? Wait, no, let's think again.

Wait, the two triangles are similar. Let's denote the first triangle with legs 43 and $x$, and the second triangle with legs 16 and 32. Since they are similar, the ratios of corresponding legs are equal. So $\frac{43}{16}=\frac{x}{32}$? No, that would be if the vertical leg of the first is $x$ and the vertical leg of the second is 32, and horizontal leg of the first is 43 and horizontal leg of the second is 16. Wait, cross - multiplying: $16x=43\times32$. Wait, no, maybe the correct proportion is $\frac{43}{x}=\frac{16}{32}$. Wait, let's solve for $x$.

Wait, if the triangles are similar, then $\frac{\text{horizontal side of first triangle}}{\text{vertical side of first triangle}}=\frac{\text{horizontal side of second triangle}}{\text{vertical side of second triangle}}$. So $\frac{43}{x}=\frac{16}{32}$. Cross - multiply: $16x = 43\times32$. Then $x=\frac{43\times32}{16}$.

Step2: Simplify the expression

Simplify $\frac{43\times32}{16}$. We know that $\frac{32}{16} = 2$. So $x=43\times2=86$? Wait, no, that can't be. Wait, maybe the proportion is $\frac{43}{16}=\frac{x + 32}{32}$? No, let's look at the diagram again. Wait, the vertical segment is $x+32$? No, the diagram shows that the vertical side of the first triangle is $x$, and the vertical side of the second triangle is 32, and the horizontal side of the first is 43, horizontal side of the second is 16. Since the triangles are similar, $\frac{43}{16}=\frac{x}{32}$. Then $x=\frac{43\times32}{16}=43\times2 = 86$? Wait, but let's check.

Wait, another way: The two triangles are similar. So the ratio of corresponding sides. Let's consider the hypotenuse - leg or leg - leg similarity. Let's set up the proportion as $\frac{43}{16}=\frac{x}{32}$. Cross - multiply: $16x=43\times32$. Then $x=\frac{43\times32}{16}$. Since $32\div16 = 2$, then $x = 43\times2=86$? Wait, but that seems large. Wait, maybe the proportion is $\frac{43}{x}=\frac{16}{32}$. Then $16x = 43\times32$, so $x=\frac{43\times32}{16}=86$. Yes, that's correct.

Step1 (Correct): Set up the proportion for similar triangles

Since the two right - angled triangles are similar (AA similarity: right angle and vertical angles are equal), the ratios of corresponding sides are equal. Let the legs of the first triangle be 43 (horizontal) and $x$ (vertical), and the legs of the second triangle be 16 (horizontal) and 32 (vertical). Then $\frac{43}{16}=\frac{x}{32}$.

Step2: Solve for $x$

Cross - multiply the proportion $\frac{43}{16}=\frac{x}{32}$:

$$16x=43\times32$$

Simplify the right - hand side: $43\times32 = 1376$
Then divide both sides by 16:

$$x=\frac{1376}{16}$$

Since $1376\div16…

Answer:

86