Question

use similar triangles and a proportion to find the length of the lake shown here. (hint: the side 76 m long in the smaller triangle corresponds to side of 76 m + 96 m = 172 m in the larger triangle.) n = \square m

Explanation:

Step1: Set up proportion for similar triangles

Since the triangles are similar, the ratios of corresponding sides are equal. Let the length of the lake be \( n \). The ratio of the base of the smaller triangle to the base of the larger triangle is equal to the ratio of the height of the smaller triangle to the height of the larger triangle. So, \(\frac{76}{172}=\frac{38}{n}\) (Wait, no, actually, the height of the smaller triangle is 38 m, and the height of the larger triangle is \( 38 + n \)? Wait, no, looking at the diagram, the smaller triangle has base 76 m and height 38 m, the larger triangle has base 172 m (76 + 96) and height \( 38 + n \)? Wait, no, maybe I misread. Wait, the hint says the 76 m side in the smaller triangle corresponds to 172 m in the larger triangle. And the height of the smaller triangle is 38 m, and the height of the larger triangle is \( 38 + n \)? No, maybe the two triangles are similar, so the ratio of corresponding sides: the base of small is 76, base of large is 172, and the height of small is 38, height of large is \( 38 + n \)? Wait, no, maybe the vertical side of the small triangle is 38, and the vertical side of the large triangle is \( n \), and the horizontal side of small is 76, horizontal side of large is 172. Wait, that makes more sense. So similar triangles, so \(\frac{76}{172}=\frac{38}{n}\)? No, that would be if 38 corresponds to \( n \), but 76 corresponds to 172. Wait, no, let's correct. If the smaller triangle has base 76 and height 38, and the larger triangle has base 172 (76 + 96) and height \( 38 + n \)? No, maybe the two triangles are such that the smaller one is inside the larger one, with the base of the small being 76, and the base of the large being 172, and the height of the small is 38, and the height of the large is \( n \). Wait, the diagram shows a small triangle with base 76, height 38, and a larger triangle with base 172, and the height is \( 38 + n \)? No, maybe the vertical segment of the large triangle is \( n \), and the small triangle's vertical is 38, horizontal is 76, large triangle's horizontal is 172, vertical is \( n \). Wait, no, the correct proportion is \(\frac{\text{base of small}}{\text{base of large}}=\frac{\text{height of small}}{\text{height of large}}\). Wait, the base of small is 76, base of large is 172, height of small is 38, height of large is \( 38 + n \)? No, that can't be. Wait, maybe the height of the large triangle is \( n \), and the height of the small is 38, base of small is 76, base of large is 172. So \(\frac{76}{172}=\frac{38}{n}\)? Solving that: cross - multiply, \( 76n = 172\times38 \). Let's calculate \( 172\times38=(170 + 2)\times38 = 170\times38+2\times38 = 6460+76 = 6536 \). Then \( n=\frac{6536}{76}=86 \)? Wait, no, that's not right. Wait, maybe I mixed up the correspondence. The correct correspondence: the side of length 76 in the small triangle corresponds to 172 in the large triangle, and the side of length 38 in the small triangle corresponds to \( n \) in the large triangle? No, wait, maybe the two triangles are similar, so the ratio of the sides: \(\frac{76}{172}=\frac{38}{n}\) is wrong. Wait, let's think again. The smaller triangle has a horizontal side of 76 m and a vertical side of 38 m. The larger triangle has a horizontal side of 172 m (76 + 96) and a vertical side of \( 38 + n \)? No, that doesn't make sense. Wait, maybe the vertical side of the large triangle is \( n \), and the vertical side of the small is 38, and the horizontal side of the small is 76, horizontal side of the large is 172. So the ratio of horizontal sides is 76/172, and the ratio of vertical sides is 38/n. So \(\frac{76}{172}=\frac{38}{n}\). Wait, but 76 is twice 38 (76 = 2×38), and 172 is twice 86 (172 = 2×86). So if 76 corresponds to 172, and 38 corresponds to n, then since 76 is 2×38, 172 should be 2×n, so n = 172/2 = 86? Wait, no, 172 divided by 2 is 86, but 38×2 is 76. Wait, maybe the correct proportion is \(\frac{76}{172}=\frac{38}{n}\), but solving that: 76n = 172×38. 172×38 = 6536, 6536÷76 = 86. Wait, but let's check the ratio: 76/172 = 19/43, 38/86 = 19/43. Yes, that's correct. Wait, but maybe the other way: the ratio of large to small is 172/76 = 2.263, and 38×2.263≈86. So n = 86? Wait, no, wait, maybe the height of the large triangle is \( 38 + n \), and the base is 172, and the small triangle has base 76 and height 38. So \(\frac{76}{172}=\frac{38}{38 + n}\). Then cross - multiply: 76(38 + n)=172×38. 76×38+76n = 172×38. 76n=(172 - 76)×38. 76n = 96×38. 76n = 3648. n = 3648÷76 = 48. Wait, now I'm confused. Wait, the hint says "the side 76 m long in the smaller triangle corresponds to side of 76 m + 96 m = 172 m in the larger triangle". So the 76 m side (horizontal) of the small triangle corresponds to 172 m (horizontal) of the large triangle. The vertical side of the small triangle is 38 m, and the vertical side of the large triangle is \( 38 + n \)? No, the lake is the vertical segment, so the small triangle has height 38, and the large triangle has height \( 38 + n \), and base 172, small base 76. So similar triangles, so \(\frac{76}{172}=\frac{38}{38 + n}\). Let's solve that:

76(38 + n)=172×38

2888+76n = 6536

76n = 6536 - 2888

76n = 3648

n = 3648÷76 = 48. Ah, that makes sense. Because 76×48 = 3648, and 172×38 = 6536, 6536 - 2888 = 3648. So the mistake earlier was in the correspondence of the height. The height of the small triangle is 38, and the height of the large triangle is 38 + n, so the ratio of bases (76/172) equals the ratio of heights (38/(38 + n)).

Step2: Solve the proportion

Starting with \(\frac{76}{172}=\frac{38}{38 + n}\)

Cross - multiply: \( 76\times(38 + n)=172\times38 \)

Calculate \( 172\times38 = 6536 \) and \( 76\times38 = 2888 \)

So \( 2888+76n = 6536 \)

Subtract 2888 from both sides: \( 76n = 6536 - 2888 = 3648 \)

Divide both sides by 76: \( n=\frac{3648}{76}=48 \)

Answer:

48