Question

12/09 - similarity word problems - l
the drawing below can be used to find x, the width of pearl pong at its widest point. what is the width of the pond?
image of two similar triangles, one small with base 12 and height 4, one large with side 38 and base x (pearl pond)
the width of the pond is \boxed{}

Explanation:

Step1: Identify similar triangles

The two triangles (small and large) are similar by AA (Angle-Angle) similarity (right angle and common angle). So, their corresponding sides are proportional.
Let the height of the small triangle be \( 4 \) and the height of the large triangle be \( 4 + 38=42 \)? Wait, no, wait. Wait, the small triangle has height \( 4 \), and the large triangle (including the small one) has a side of length \( 38 \)? Wait, maybe the small triangle has base \( 12 \), height \( 4 \), and the large triangle has height \( 4 + 38 \)? No, maybe the distance from the vertex to the pond's edge is \( 38 \), and the small triangle's height is \( 4 \). So the ratio of heights is \( \frac{4 + 38}{4}=\frac{42}{4}=\frac{21}{2} \)? Wait, no, maybe the small triangle has height \( 4 \), and the large triangle (the one with the pond) has a side from the vertex to the pond's side as \( 38 \), so the total height for the large triangle is \( 4 + 38 \)? Wait, no, perhaps the small triangle has base \( 12 \), height \( 4 \), and the large triangle (the one containing the pond) has a height of \( 4 + 38 = 42 \)? Wait, no, maybe the small triangle's height is \( 4 \), and the large triangle's height is \( 38 \)? No, that doesn't make sense. Wait, the correct approach: similar triangles, so \( \frac{\text{base of small triangle}}{\text{base of large triangle}}=\frac{\text{height of small triangle}}{\text{height of large triangle}} \). Let's assume the small triangle has base \( 12 \), height \( 4 \), and the large triangle has height \( 4 + 38 = 42 \)? Wait, no, maybe the distance from the vertex to the pond is \( 38 \), so the height of the large triangle (from vertex to pond's base) is \( 38 \), and the small triangle's height is \( 4 \)? No, that can't be. Wait, maybe the small triangle is above the pond, with height \( 4 \), base \( 12 \), and the large triangle (the one including the pond) has a height of \( 4 + 38 = 42 \)? Wait, no, let's re-express. Let the height of the small triangle be \( h_1 = 4 \), base \( b_1 = 12 \). The height of the large triangle (the one with the pond) is \( h_2 = 4 + 38 = 42 \)? Wait, no, maybe the height of the large triangle is \( 38 \), and the small triangle's height is \( 4 \)? No, that would be \( \frac{12}{x}=\frac{4}{38 + 4} \)? Wait, no, perhaps the small triangle has height \( 4 \), and the large triangle (the one from the vertex to the pond's base) has height \( 4 + 38 = 42 \). So the ratio of bases is equal to the ratio of heights. So \( \frac{12}{x}=\frac{4}{4 + 38} \)? Wait, no, \( \frac{12}{x}=\frac{4}{38 + 4} \)? Wait, \( 4 + 38 = 42 \), so \( \frac{12}{x}=\frac{4}{42} \)? No, that would give \( x=\frac{12\times42}{4}=126 \). Wait, but let's check again. Wait, maybe the height of the small triangle is \( 4 \), and the height of the large triangle (the one with the pond) is \( 38 \). Then \( \frac{12}{x}=\frac{4}{38} \), so \( x=\frac{12\times38}{4}=114 \). Wait, that makes more sense. Oh! I see, maybe the small triangle has height \( 4 \), and the large triangle (the one containing the pond) has a height of \( 38 \). Wait, no, the distance from the vertex to the pond's edge is \( 38 \), so the height of the large triangle (from vertex to pond's base) is \( 38 \), and the small triangle's height is \( 4 \). So the ratio of bases is \( \frac{12}{x}=\frac{4}{38} \)? No, that would be if the heights are \( 4 \) and \( 38 \). Wait, no, the correct proportion is \( \frac{\text{base of small}}{\text{base of large}}=\frac{\text{height of small}}{\text{height of large}} \). Let's define:

Let \( h_1 = 4 \) (height of small triangle), \( b_1 = 12 \) (base of small triangle).

\( h_2 = 4 + 38 = 42 \) (height of large triangle, from vertex to pond's base).

Then \( \frac{b_1}{b_2}=\frac{h_1}{h_2} \)

So \( \frac{12}{x}=\frac{4}{42} \)

Wait, no, \( h_2 \) should be the height of the large triangle, which is \( 4 + 38 = 42 \), so \( \frac{12}{x}=\frac{4}{42} \)

Then \( x=\frac{12\times42}{4}=126 \). Wait, but that seems high. Alternatively, maybe the height of the large triangle is \( 38 \), and the small triangle's height is \( 4 \), so \( \frac{12}{x}=\frac{4}{38} \), so \( x=\frac{12\times38}{4}=114 \). Wait, maybe I misread the diagram. Let's assume that the small triangle has a height of \( 4 \), and the distance from the small triangle to the pond is \( 38 \), so the total height for the large triangle (from the vertex to the pond's base) is \( 4 + 38 = 42 \). Then the ratio of bases is equal to the ratio of heights. So \( \frac{12}{x}=\frac{4}{42} \), so \( x = \frac{12 \times 42}{4} = 126 \). Wait, but let's check the calculation: \( 12 \times 42 = 504 \), \( 504 \div 4 = 126 \). Alternatively, if the height of the large triangle is \( 38 \), then \( 12 \times 38 = 456 \), \( 456 \div 4 = 114 \). Which is correct? Let's think about similar triangles. The two triangles are similar, so corresponding sides are proportional. The small triangle has base \( 12 \), height \( 4 \). The large triangle (the one with the pond) has height \( 4 + 38 = 42 \) (since the distance from the small triangle to the pond is \( 38 \), so total height is \( 4 + 38 \)). So the ratio of heights is \( \frac{4}{42} = \frac{2}{21} \), so the ratio of bases is also \( \frac{2}{21} \). So \( \frac{12}{x} = \frac{2}{21} \), so \( x = \frac{12 \times 21}{2} = 126 \). Yes, that makes sense. So:

Step1: Set up proportion for similar triangles

Since the triangles are similar, \( \frac{\text{Base of small triangle}}{\text{Base of large triangle}}=\frac{\text{Height of small triangle}}{\text{Height of large triangle}} \)

Let \( x \) be the width of the pond (base of large triangle), height of small triangle \( = 4 \), height of large triangle \( = 4 + 38 = 42 \), base of small triangle \( = 12 \)

So \( \frac{12}{x}=\frac{4}{42} \)

Step2: Solve for \( x \)

Cross - multiply: \( 4x = 12\times42 \)

\( 4x = 504 \)

Divide both sides by \( 4 \): \( x=\frac{504}{4}=126 \)

Answer:

\( 126 \)