Question

what is the area of the shaded region?
29 mm
17 mm
16 mm
30 mm
write your answer as a whole number or
a decimal rounded to the nearest
hundredth.
square millimeters

Explanation:

Step1: Find area of large triangle

The large triangle has legs 29 mm and 30 mm. Area of triangle is $\frac{1}{2} \times base \times height$. So, $\frac{1}{2} \times 29 \times 30 = 435$ square mm.

Step2: Find area of small triangle

The small triangle: check if it's right - angled (16,17,? Let's see $16^{2}+17^{2}=256 + 289 = 545$, wait no, maybe it's a right triangle? Wait, no, wait the large triangle is right - angled (since legs 29 and 30, and the small triangle inside: let's check if 16,17 and the other side. Wait, actually, the small triangle: let's assume it's a right triangle? Wait, no, wait the large triangle is right - angled (because it's a triangle with legs 29 and 30, so right - angled at the right angle). The small triangle: let's calculate its area. Wait, maybe the small triangle has base 16 and height? Wait, no, maybe we can use Heron's formula? Wait, no, wait 16,17: wait, $16^{2}+15^{2}=256 + 225 = 481$, no. Wait, maybe it's a right triangle? Wait, no, maybe I made a mistake. Wait, the large triangle: area is $\frac{1}{2}\times29\times30 = 435$. The small triangle: let's check if it's a right triangle. Wait, 16 and 15? Wait, no, maybe the small triangle has base 16 and height such that it's a right triangle. Wait, wait, 17 is the hypotenuse? So, if hypotenuse is 17, one leg 16, then the other leg is $\sqrt{17^{2}-16^{2}}=\sqrt{289 - 256}=\sqrt{33}\approx5.7446$. Then area of small triangle is $\frac{1}{2}\times16\times\sqrt{33}\approx\frac{1}{2}\times16\times5.7446 = 8\times5.7446\approx45.9568$. Wait, no, that can't be. Wait, maybe the large triangle is right - angled, and the small triangle is also right - angled, with legs 16 and let's say h, and hypotenuse 17. So, $h=\sqrt{17^{2}-16^{2}}=\sqrt{33}\approx5.7446$. Then area of small triangle is $\frac{1}{2}\times16\times\sqrt{33}\approx45.96$. Then area of shaded region is large area - small area: $435 - 45.96 = 389.04$? Wait, no, that doesn't seem right. Wait, maybe the large triangle has legs 29 and 30, so area is $\frac{29\times30}{2}=435$. The small triangle: maybe it's a right triangle with legs 16 and 15? Wait, 15 - 16 - 17 is a Pythagorean triple? Wait, $15^{2}+16^{2}=225 + 256 = 481$, $17^{2}=289$, no. Wait, 8 - 15 - 17: $8^{2}+15^{2}=64 + 225 = 289 = 17^{2}$. Oh! Wait, I made a mistake. The small triangle: if hypotenuse is 17, and one leg is 8, the other is 15? But in the diagram, one side is 16. Wait, no, maybe the diagram has the small triangle with base 16 and the other leg is 15? Wait, no, the diagram shows 17 mm (hypotenuse), 16 mm (one leg). So, the other leg is $\sqrt{17^{2}-16^{2}}=\sqrt{289 - 256}=\sqrt{33}\approx5.7446$. Then area of small triangle is $\frac{1}{2}\times16\times\sqrt{33}\approx45.96$. Then shaded area is $435 - 45.96 = 389.04$. Wait, but maybe the large triangle is not right - angled? No, the diagram shows a right - angled triangle (since it's a triangle with two legs 29 and 30, so right - angled at the vertex where 29 and 30 meet). Wait, maybe I misread the diagram. Wait, the large triangle: vertical side 29, horizontal side 30, so right - angled. The small triangle: inside, with one side 17 (hypotenuse), one side 16 (horizontal), so vertical side is $\sqrt{17^{2}-16^{2}}=\sqrt{33}\approx5.7446$. Then area of small triangle: $\frac{1}{2}\times16\times\sqrt{33}\approx45.96$. Area of large triangle: $\frac{1}{2}\times29\times30 = 435$. Then shaded area: $435 - 45.96 = 389.04$. Wait, but maybe the small triangle is a right triangle with legs 16 and 15? Wait, 15 - 16 - 17 is not a Pythagorean triple. 8 - 15 - 17 is. So maybe the small triangle has legs 8 and 15, and hypotenuse 17, and the base is 16? No, that doesn't match. Wait, maybe the diagram has the small triangle with base 16 and height 15? Then area would be $\frac{1}{2}\times16\times15 = 120$. Then shaded area would be $435 - 120 = 315$. But why? Because 8 - 15 - 17: 8 and 15 are legs, 17 hypotenuse. Maybe the diagram has the small triangle with leg 15 (vertical) and 16 (horizontal)? But 15 and 16: $15^{2}+16^{2}=481$, 17^{2}=289, no. Wait, I think I made a mistake in assuming the large triangle's legs. Wait, the large triangle: vertical side 29, horizontal side 30, so it's a right triangle. The small triangle: let's check if it's a right triangle. If the small triangle has sides 16, 17, and x, and if it's a right triangle, then either 16^{2}+x^{2}=17^{2} (so x = $\sqrt{33}$) or 17^{2}+16^{2}=x^{2} (which is bigger). So the first case. Then area of small triangle is $\frac{1}{2}\times16\times\sqrt{33}\approx45.96$. Then shaded area is 435 - 45.96 = 389.04. But maybe the problem is that the large triangle is a right triangle, and the small triangle is also a right triangle, and the vertical side of the large triangle is 29, horizontal 30, and the small triangle has horizontal side 16, and vertical side equal to (29 - something)? No, the diagram shows the small triangle inside, with one side 17 (the hypotenuse), one side 16 (the base), and the other side vertical. So the vertical side of the small triangle is $\sqrt{17^{2}-16^{2}}=\sqrt{33}\approx5.7446$. Then area of small triangle is $\frac{1}{2}\times16\times\sqrt{33}\approx45.96$. Then shaded area is $435 - 45.96 = 389.04$. Wait, but maybe the large triangle is not right - angled? No, the way it's drawn, with a right angle at the bottom left, so vertical and horizontal sides are legs. So I think that's the way.
Wait, another approach: maybe the large triangle is a right triangle with legs 29 and 30, area = (29*30)/2 = 435. The small triangle: let's check if it's a right triangle. If 16 and 15 are legs, area = (16*15)/2 = 120. But 15 - 16 - 17: 15²+16²=225 + 256 = 481, 17²=289, not equal. So that's wrong. Wait, 8 - 15 - 17: 8²+15²=64 + 225 = 289 = 17². So if the small triangle has legs 8 and 15, hypotenuse 17, and the base is 8? But the diagram shows base 16. Oh! Wait, maybe the small triangle is similar to the large triangle? Let's check the ratios. Large triangle legs: 29 and 30. Small triangle: 16 and x, hypotenuse 17. If similar, then 29/16 = 30/x = 29/17? No, that doesn't make sense. Wait, maybe the large triangle is a right triangle, and the small triangle is also a right triangle, and the area of the shaded region is area of large right triangle minus area of small right triangle. So large area: (29*30)/2 = 435. Small area: let's find the legs of the small right triangle. Given hypotenuse 17, one leg 16, so the other leg is $\sqrt{17^{2}-16^{2}}=\sqrt{289 - 256}=\sqrt{33}\approx5.7446$. Then small area: (16*$\sqrt{33}$)/2 = 8*$\sqrt{33}\approx8*5.7446\approx45.96$. Then shaded area: 435 - 45.96 = 389.04. So the answer is approximately 389.04.

Answer:

389.04