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 13 inches, and the length of the base is 8 inches. Find the triangle's perimeter. Round to the nearest tenth of an inch. Answer Attemptiout of 5 square in

## Step 1<br />The problem involves an isosceles triangle, which is a triangle with two equal sides. The altitude drawn from the vertex of the isosceles triangle forms two right-angled triangles. The altitude is the height of these right-angled triangles, and the base of the isosceles triangle is divided into two equal segments by the altitude.<br /><br />## Step 2<br />We can use the Pythagorean theorem to find the length of the equal sides of the isosceles triangle. The Pythagorean theorem states that in a right-angled triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides. This can be written as:<br />### \(a^2 + b^2 = c^2\)<br />where \(a\) and \(b\) are the lengths of the two legs of the right-angled triangle, and \(c\) is the length of the hypotenuse.<br /><br />## Step 3<br />In this case, the altitude is one of the legs of the right-angled triangle, and half of the base of the isosceles triangle is the other leg. The equal side of the isosceles triangle is the hypotenuse. So, we can substitute the given values into the Pythagorean theorem:<br />### \(a = 13\) (altitude)<br />### \(b = \frac{8}{2} = 4\) (half of the base)<br />### \(c =?\) (the equal side of the isosceles triangle)<br /><br />## Step 4<br />Substituting these values into the Pythagorean theorem gives:<br />### \(c = \sqrt{a^2 + b^2} = \sqrt{13^2 + 4^2} = \sqrt{169 + 16} = \sqrt{185}\)<br /><br />## Step 5<br />The perimeter of the isosceles triangle is the sum of the lengths of all its sides. Since the triangle is isosceles, two of its sides are equal. Therefore, the perimeter \(P\) is given by:<br />### \(P = 2c + b\)<br /><br />## Step 6<br />Substituting the values of \(c\) and \(b\) into the formula for the perimeter gives:<br />### \(P = 2\sqrt{185} + 4\)<br /><br />## Step 7<br />Finally, we round the perimeter to the nearest tenth of an inch.