Question

what is the area of the shaded region? 40 in 26 in 18 in 27 in write your answer as a whole number or a decimal rounded to the nearest hundredth.

Explanation:

Step1: Calculate area of large triangle

The formula for the area of a triangle is $A = \frac{1}{2} \times base \times height$. For the large triangle, base = 27 in, height = 40 in.
$A_{large} = \frac{1}{2} \times 27 \times 40 = 540$ square inches.

Step2: Calculate area of small triangle

For the small triangle, base = 18 in, height can be found using Pythagoras? Wait, no, wait. Wait, the small triangle: wait, actually, the small triangle has sides 18, 26? Wait, no, wait, maybe it's a right triangle? Wait, 18-26-? Wait, no, wait, the large triangle is a right triangle (since the height is 40 and base 27, and the small triangle inside: wait, the small triangle has base 18 and hypotenuse 26? Wait, no, maybe the small triangle is also a right triangle. Let's check: if base is 18, hypotenuse 26, then height (let's call it h) is $\sqrt{26^2 - 18^2} = \sqrt{676 - 324} = \sqrt{352} \approx 18.76$? Wait, no, that can't be. Wait, maybe the small triangle is a right triangle with base 18 and height corresponding? Wait, no, maybe I made a mistake. Wait, the large triangle: base 27, height 40, so area is 0.5*27*40=540. The small triangle: base 18, and what's the height? Wait, maybe the small triangle is similar? Wait, no, the figure shows a large right triangle with a smaller right triangle inside, so the shaded area is large triangle area minus small triangle area. Wait, the small triangle: let's check if it's a right triangle. If the small triangle has base 18 and hypotenuse 26, then the height (vertical side) is $\sqrt{26^2 - 18^2} = \sqrt{676 - 324} = \sqrt{352} \approx 18.76$? No, that doesn't make sense. Wait, maybe the small triangle is a right triangle with legs 18 and let's say h, and hypotenuse 26. Then $18^2 + h^2 = 26^2$ → $h^2 = 676 - 324 = 352$ → $h = \sqrt{352} ≈ 18.76$. But the large triangle has height 40, so that can't be. Wait, maybe the small triangle's height is proportional? Wait, no, maybe the large triangle is a right triangle with base 27 and height 40, so area 540. The small triangle: base 18, and height: wait, maybe the small triangle is also a right triangle, and the shaded area is the difference. Wait, maybe I miscalculated. Wait, let's re-express:

Wait, the large triangle: area = 0.5 * 27 * 40 = 540.

Small triangle: let's check if it's a right triangle. If the small triangle has base 18 and hypotenuse 26, then the other leg (height) is $\sqrt{26^2 - 18^2} = \sqrt{676 - 324} = \sqrt{352} ≈ 18.76$. But that seems odd. Wait, maybe the small triangle is a right triangle with legs 18 and 24? Wait, 18-24-30, but 26 is given. Wait, no, the problem says the small triangle has sides 18 and 26? Wait, the figure: the small triangle has base 18 in, and the other side (the vertical side) is 26? No, the label 26 in is inside the small triangle, maybe the hypotenuse. Wait, maybe the small triangle is a right triangle, so area is 0.5 * 18 * (height). Wait, maybe I made a mistake. Wait, let's calculate the area of the small triangle. If it's a right triangle, then area is 0.5 * base * height. Wait, maybe the small triangle has base 18 and height 24? Wait, 18-24-30, but 26 is given. Wait, no, 18^2 + 24^2 = 324 + 576 = 900 = 30^2, not 26. So that's not. Wait, 26^2 - 18^2 = 676 - 324 = 352, so height is sqrt(352) ≈ 18.76. Then area of small triangle is 0.5 * 18 * 18.76 ≈ 0.5 * 337.68 ≈ 168.84. Then shaded area is 540 - 168.84 ≈ 371.16? Wait, but that seems off. Wait, maybe the large triangle is a right triangle, and the small triangle is also a right triangle, and the sides are proportional? Wait, 27 and 40: 27/18 = 1.5, 40/26 ≈ 1.538, not proportional. Wait, maybe I misread the figure. Wait, the large triangle: base 27, height 40. The small triangle: base 18, and the other side (the vertical side) is 26? No, maybe the small triangle's height is 26? Wait, no, the label 26 is inside the small triangle, maybe the height. Wait, if the small triangle has base 18 and height 26, then area is 0.5 * 18 * 26 = 234. Then shaded area is 540 - 234 = 306. But that would be if the small triangle is a right triangle with base 18 and height 26. Wait, but then the hypotenuse would be sqrt(18^2 + 26^2) = sqrt(324 + 676) = sqrt(1000) ≈ 31.62, but the figure doesn't show that. Wait, maybe the large triangle is a right triangle, and the small triangle is also a right triangle, so the shaded area is large area minus small area. Let's check the two possibilities:

Case 1: Small triangle has base 18, height 26 (right triangle). Then area = 0.5*18*26 = 234. Large area = 0.5*27*40 = 540. Shaded area = 540 - 234 = 306.

Case 2: Small triangle has hypotenuse 26, base 18. Then height = sqrt(26^2 - 18^2) = sqrt(352) ≈ 18.76. Area = 0.5*18*18.76 ≈ 168.84. Shaded area = 540 - 168.84 ≈ 371.16.

But which is correct? Wait, the large triangle: base 27, height 40. Let's check if 27-40-? Hypotenuse would be sqrt(27^2 + 40^2) = sqrt(729 + 1600) = sqrt(2329) ≈ 48.26. The small triangle: if base 18, height 26, hypotenuse sqrt(18^2 + 26^2) = sqrt(1000) ≈ 31.62. Not similar. If the small triangle's height is proportional: 40 / h = 27 / 18 → h = (40*18)/27 = 80/3 ≈ 26.67. But the label is 26, close to 26.67. Maybe a typo, or maybe the small triangle has height 26. Let's go with the first case, assuming the small triangle is a right triangle with base 18 and height 26. Then:

Step1: Area of large triangle

$A_{large} = \frac{1}{2} \times 27 \times 40 = 540$

Step2: Area of small triangle

$A_{small} = \frac{1}{2} \times 18 \times 26 = 234$

Step3: Shaded area

$A_{shaded} = A_{large} - A_{small} = 540 - 234 = 306$

Wait, but let's check the other way. If the small triangle is a right triangle with hypotenuse 26 and base 18, then height is sqrt(26² - 18²) = sqrt(676 - 324) = sqrt(352) ≈ 18.76. Then area is 0.5*18*18.76 ≈ 168.84. Then shaded area is 540 - 168.84 ≈ 371.16. But which is correct? Wait, the large triangle: 27 and 40, so area 540. The small triangle: if the vertical side is 26, then base 18, area 234. Then shaded is 306. Alternatively, maybe the small triangle is similar. Let's check similarity: 27/18 = 1.5, 40/h = 1.5 → h = 40/1.5 ≈ 26.67. Close to 26, maybe rounded. So 26.67 is 80/3. Then area of small triangle would be 0.5*18*(80/3) = 0.5*18*26.666... = 9*26.666... = 240. Then shaded area is 540 - 240 = 300. But the problem says 26 in. Maybe the small triangle has height 26, so area 234, shaded 306. Let's go with that.

Answer:

306