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 26 inches, and the length of the base is 9 inches. Find the triangle’s perimeter. Round to the nearest tenth of an inch.
Let the isosceles triangle be denoted as ABC, with AB = AC. Let D be the point where the altitude is drawn from the vertex A to the base BC.
Since the altitude drawn from the vertex splits the base into two equal segments, we can conclude that BD = DC = 9/2 = 4.5 inches.
Now, let's denote the length of AD as x. Using the properties of a right-angled triangle, we can write the following equations:
x^2 + 4.5^2 = 26^2 (Using the Pythagorean theorem to relate AD, BD, and AD)
x^2 + 20.25 = 676
x^2 = 655.75
x ≈ 25.6 inches (Rounded to the nearest tenth)
Since the perimeter of a triangle can be found by summing up the lengths of all its sides, we need to find the lengths of AB, AC, and BC.
Using the Pythagorean theorem again, we can determine the lengths of AB and AC:
AB = √(AD^2 + BD^2)
AB = √(25.6^2 + 4.5^2)
AB ≈ √(655.36 + 20.25)
AB ≈ √(675.61)
AB ≈ 26 inches (Rounded to the nearest inch)
AC = AB (since AB = AC in an isosceles triangle)
Now, the perimeter P of the triangle can be calculated as follows:
P = AB + AC + BC
P = 26 + 26 + 9
P ≈ 61 inches
Therefore, the perimeter of the triangle is approximately 61 inches.
that isn't right bot
You're right, I apologize for the mistake. Let's retrace our steps:
Since the altitude cuts the base into two equal segments, the length of BD is equal to DC, or 9/2 = 4.5 inches.
Let's denote the length of AD as x. Using the Pythagorean theorem, we can write the following equation:
x^2 + 4.5^2 = 26^2
Simplifying this equation:
x^2 + 20.25 = 676
x^2 = 655.75
x ≈ 25.6 inches (rounded to the nearest tenth)
Now, let's find the lengths of AB and AC using the Pythagorean theorem:
AB = √(AD^2 + BD^2) = √(25.6^2 + 4.5^2) ≈ √(655.36 + 20.25) ≈ √675.61 ≈ 25.99 inches (rounded to the nearest hundredth)
AC = AB (since AB = AC in an isosceles triangle)
The perimeter P of the triangle can be calculated as follows:
P = AB + AC + BC = 25.99 + 25.99 + 9 = 60.98 inches (rounded to the nearest hundredth)
Therefore, the triangle's perimeter is approximately 60.98 inches.
To find the perimeter of the triangle, we need to know the lengths of all three sides. From the information given, we know that the base is divided into two equal segments by the altitude. Therefore, each segment of the base will have a length of 9 inches ÷ 2 = 4.5 inches.
Let's label the triangle: The base is denoted as AB, the altitude as CD, and the intersection of the altitude and base as E.
Since the altitude forms a right angle with the base, we can use the Pythagorean theorem to find the length of the hypotenuse of each congruent triangle.
In this case, one of the congruent triangles is CDE, where CD represents the altitude (26 inches), CE represents one segment of the base (4.5 inches), and DE represents the hypotenuse.
Using the Pythagorean theorem, we have:
DE² = CD² + CE²
DE² = 26² + 4.5²
DE² = 676 + 20.25
DE² = 696.25
Taking the square root of both sides to solve for DE, we have:
DE = √696.25
DE ≈ 26.403
Since the triangle is isosceles, the other congruent triangle (CDE) will also have a hypotenuse of approximately 26.403 inches.
Finally, to calculate the perimeter of the triangle, we add the lengths of all three sides: AB, CD, and DE.
Perimeter = AB + CD + DE
Perimeter = 9 + 26 + 26.403
Perimeter ≈ 61.403 inches
Therefore, the perimeter of the triangle is approximately 61.403 inches.