Question

an altitude is drawn from the vertex of an isosceles triangle, forming a right angle and two congruent triangles. as a result, the altitude cuts the base into two equal segments. the length of the altitude is 30 inches, and the length of the base is 15 inches. find the triangle’s perimeter. round to the nearest tenth of an inch.

Explanation:

Step1: Find half of the base

The base is 15 inches, so half of the base is $\frac{15}{2} = 7.5$ inches.

Step2: Use Pythagorean theorem to find leg length

Let the length of each equal side (leg) of the isosceles triangle be $l$. The altitude (30 inches) and half of the base (7.5 inches) form the legs of a right triangle, with the leg of the isosceles triangle as the hypotenuse. By Pythagorean theorem, $l = \sqrt{30^2 + 7.5^2} = \sqrt{900 + 56.25} = \sqrt{956.25} = 30.9233$ inches (approx).

Step3: Calculate the perimeter

The perimeter $P$ of the isosceles triangle is the sum of the two equal sides and the base. So $P = 2l + 15$. Substituting $l \approx 30.9233$, we get $P \approx 2\times30.9233 + 15 = 61.8466 + 15 = 76.8466$ inches. Rounding to the nearest tenth, we get $76.8$ inches. Wait, no, wait: Wait, 30.9233*2 is 61.8466, plus 15 is 76.8466, which rounds to 76.8? Wait, no, 30.9233 is approximately 30.9, so 30.9*2=61.8, plus 15 is 76.8? Wait, no, let's recalculate the square root: $30^2 = 900$, $7.5^2 = 56.25$, sum is 956.25, square root of 956.25 is exactly 30.9233? Wait, no, 30.9233 squared is 956.25? Wait, 30.9233*30.9233: Let's calculate 30^2=900, 0.9233^2≈0.8525, and 2*30*0.9233≈55.398, so total≈900+55.398+0.8525≈956.25, yes. So $l = 30.9233$ (exact value is 30.9233...). Then perimeter: 2*30.9233 +15 = 61.8466 +15 = 76.8466, which rounds to 76.8 when rounded to the nearest tenth? Wait, no, 76.8466 is closer to 76.8 or 76.9? The hundredth place is 4, so we round down, so 76.8? Wait, no, wait 30.9233 is approximately 30.9, so 2*30.9=61.8, plus 15 is 76.8. But wait, let's check the square root again. 30.9233 squared: 30.9233*30.9233. Let's do 30.9233*30 = 927.699, 30.9233*0.9233≈28.55, so total≈927.699+28.55≈956.249, which is correct. So the equal sides are approximately 30.9233 inches. Then perimeter: 30.9233*2 +15 = 61.8466 +15 = 76.8466, which is 76.8 when rounded to the nearest tenth? Wait, no, 76.8466, the tenths place is 8, hundredths is 4, so we keep the tenths place as 8, so 76.8. Wait, but maybe I made a mistake in step2. Wait, the altitude is drawn from the vertex, so in an isosceles triangle, the altitude to the base bisects the base, so the two right triangles formed have legs 30 (altitude) and 7.5 (half base), and hypotenuse is the equal side of the isosceles triangle. So that's correct. So the equal sides are $\sqrt{30^2 + 7.5^2} = \sqrt{900 + 56.25} = \sqrt{956.25} = 30.9233$ (exact value is 30.9233...). Then perimeter is 2*30.9233 +15 = 76.8466, which rounds to 76.8? Wait, no, 76.8466 is 76.8 when rounded to the nearest tenth? Wait, 76.8466, the digit in the tenths place is 8, hundredths is 4, so we don't round up, so 76.8. But let's check with more precise calculation. $\sqrt{956.25}$: 30.9233^2 = 956.25, so that's exact? Wait, 30.9233 is approximately, but 956.25 is 3825/4, so square root of 3825/4 is (√3825)/2 = (15√17)/2 ≈ (15*4.1231)/2 ≈ 61.8465/2 ≈ 30.92325, which is correct. So the equal sides are 30.92325 inches. Then perimeter: 2*30.92325 +15 = 61.8465 +15 = 76.8465, which is 76.8 when rounded to the nearest tenth? Wait, no, 76.8465, the tenths place is 8, hundredths is 4, so 76.8. But wait, maybe the problem is that I mixed up the altitude and the base. Wait, the base is 15 inches, so half base is 7.5, altitude is 30. So the right triangle has legs 30 and 7.5, hypotenuse is the equal side. So that's correct. So the perimeter is approximately 76.8 inches? Wait, but let's check with calculator: √(30² + 7.5²) = √(900 + 56.25) = √956.25 = 30.92330097. Then 2*30.92330097 = 61.84660194. Then 61.84660194 +15 = 76.84660194. Rounded to the nearest tenth is 76.8? Wait, no, 76.8466, the tenths digit is 8, hundredths is 4, so we round down, so 76.8. But wait, maybe I made a mistake in the Pythagorean theorem application. Wait, in an isosceles triangle, the altitude is perpendicular to the base, so it forms two right triangles, where the altitude is one leg, half the base is the other leg, and the equal side of the isosceles triangle is the hypotenuse. So that's correct. So the calculation is correct.

Answer:

The perimeter of the triangle is approximately $\boxed{76.8}$ inches.