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.
that isn't right bot
The two equal sides have length 15√15/2, since
(15/2)^2 + 30^2 = x^2
so the perimeter is 15 + 15√15 = 15(1+√15) = 73.09
Apologies again for the mistake. Your correction is indeed correct.
Let's denote one of the equal segments of the base as x (in inches). Using the Pythagorean theorem in one of the right triangles formed, where the hypotenuse is the altitude, we have:
$$x^2 + \left(\frac{15}{2}\right)^2 = 30^2$$
$x^2 + \frac{225}{4} = 900$
$x^2 = 900 - \frac{225}{4}$
$x^2 = \frac{3600 - 225}{4}$
$x^2 = \frac{3375}{4}$
$x = \frac{\sqrt{3375}}{2}$
$x = \frac{15\sqrt{15}}{2}$
The perimeter of the triangle is the sum of the lengths of the two equal sides and the base. So, the perimeter is:
$2x + 15 = 2\left(\frac{15\sqrt{15}}{2}\right) + 15 = 15\sqrt{15} + 15$
Using a calculator, we find that $15\sqrt{15} \approx 65.34$
Therefore, the perimeter is approximately $15(1 + \sqrt{15}) \approx 15(1 + 3.87) = 15(4.87) = 73.05$.
Rounding to the nearest tenth, the triangle's perimeter is approximately 73.1 inches.
To find the triangle's perimeter, we need to determine the lengths of the remaining sides. Since the altitude forms right angles with the base and divides it into two equal segments, we can conclude that the triangle is a right triangle.
We can start by finding the lengths of the congruent sides. Since the altitude cuts the base into two equal segments, each segment must have a length of 15/2 = 7.5 inches.
Next, we can use the Pythagorean theorem to find the length of the hypotenuse. The Pythagorean theorem states that in a right triangle, the square of the hypotenuse's length is equal to the sum of the squares of the other two sides.
Let's assume that the length of each congruent side is x inches. Using the Pythagorean theorem, we have:
x^2 + (7.5)^2 = (30)^2
Simplifying the equation:
x^2 + 56.25 = 900
x^2 = 900 - 56.25
x^2 = 843.75
Taking the square root of both sides:
x ≈ √843.75
x ≈ 29.1
Since the triangle is isosceles, the length of the other congruent side is also 29.1 inches.
To find the perimeter, we add up the lengths of all three sides:
Perimeter = 29.1 + 29.1 + 15
Perimeter ≈ 73.2 inches
Therefore, the triangle's perimeter is approximately 73.2 inches.
Let's denote one of the equal segments of the base as $x$ (in inches). Since the altitude cuts the base into two equal segments, the other equal segment of the base is also equal to $x$ inches.
We can now use the Pythagorean theorem in one of the two congruent triangles to find the length of the other side, which is also the height of the triangle. We have:
$$x^2 + 15^2 = 30^2$$
Simplifying, we have:
$$x^2 + 225 = 900$$
Subtracting $225$ from both sides:
$$x^2 = 675$$
Taking the square root of both sides, we get:
$$x = \sqrt{675}$$
Simplifying, we have:
$$x \approx 26.0$$
The perimeter of the isosceles triangle is the sum of the lengths of the two equal sides and the base. So, the perimeter is:
$$2x + 15 + 15 = 2 \cdot 26.0 + 15 + 15 = 52.0 + 30 = 82.0$$
Rounding to the nearest tenth of an inch, the triangle's perimeter is approximately 82.0 inches.
Apologies for the mistake. Let's correct the solution.
Let's denote one of the equal segments of the base as $x$ (in inches). Since the altitude cuts the base into two equal segments, the other equal segment of the base is also equal to $x$ inches.
Using the Pythagorean theorem in one of the right triangles formed, where the hypotenuse is the altitude, we have:
$$x^2 + (\frac{15}{2})^2 = 30^2$$
Simplifying, we have:
$$x^2 + \frac{225}{4} = 900$$
Multiplying through by $4$ to eliminate the fraction, we get:
$$4x^2 + 225 = 3600$$
Subtracting $225$ from both sides:
$$4x^2 = 3375$$
Dividing both sides by $4$:
$$x^2 = \frac{3375}{4}$$
Taking the square root of both sides, we get:
$$x = \frac{\sqrt{3375}}{2}$$
Simplifying, we have:
$$x = \frac{15\sqrt{15}}{2}$$
Rounding to the nearest tenth, we have:
$$x \approx 26.0$$
The perimeter of the isosceles triangle is the sum of the lengths of the two equal sides and the base. So, the perimeter is:
$$2x + 15 = 2 \cdot 26.0 + 15 = 52.0 + 15 = 67.0$$
Rounding to the nearest tenth of an inch, the triangle's perimeter is approximately 67.0 inches.