By the Pythagorean Theorem, if \(\displaystyle a ,b\) are the legs of a right triangle and \(\displaystyle c\) is its hypotenuse, 

\(\displaystyle c^{2} = a^{2} + b^{2}\)

Substitute \(\displaystyle a = 30, b= 40\) and solve for \(\displaystyle c\):

\(\displaystyle c^{2} = 30 ^{2} + 40^{2} = 900 + 1,600 = 2,500\)

\(\displaystyle c = \sqrt{2,500} = 50\)

The perimeter of the triangle is 

\(\displaystyle P = 30 + 40 + 50 = 120\)

The hypotenuse of the right triangle can be calculated using the Pythagorean theorem:

\(\displaystyle c = \sqrt{30^{2}+40^{2}} = \sqrt{900+1600} = \sqrt{2,500} = 50\)

Add the three sides:

\(\displaystyle P = 30 + 40 + 50 = 120\)

The Pythagorean Theorem can be used to derive the length of the second leg:

\(\displaystyle b = \sqrt{50^{2}-14^{2}} = \sqrt{2,500 - 196} = \sqrt{2,304} = 48\) inches

Add the three sides to get the perimeter.

\(\displaystyle P = 14 + 48 + 50 = 112\) inches.

A triangle is right if and only if it satisfies the Pythagorean relationship

\(\displaystyle a ^{2}+ b^{2}= c^{2}\)

where \(\displaystyle c\) is the measure of the longest side and \(\displaystyle a,b\) are the other two sidelengths. We test each of the four sets of lengths, remembering to convert feet to inches by multiplying by 12.

7 inches, 2 feet, 30 inches:

2 feet is equal to 24 inches. The relationship to be tested is

\(\displaystyle 7^{2}+ 24 ^{2} = 30^{2}\)

\(\displaystyle 49 + 576 = 900\)

\(\displaystyle 625 = 900\) - False

10 inches, 1 foot, 14 inches:

1 foot is equal to 12 inches. The relationship to be tested is

\(\displaystyle 10^{2}+ 12^{2}= 14^{2}\)

\(\displaystyle 100 + 144 = 196\)

\(\displaystyle 244 = 196\) - False

15 inches, 3 feet, 40 inches:

3 feet is equal to 36 inches. The relationship to be tested is

\(\displaystyle 15^{2}+ 36^{2} = 40^{2}\)

\(\displaystyle 225 + 1,296 = 1,600\)

\(\displaystyle 1,521 = 1,600\) - False

2 feet, 32 inches, 40 inches:

2 feet is equal to 24 inches. The relationship to be tested is

\(\displaystyle 24 ^{2}+ 32^{2} = 40^{2}\)

\(\displaystyle 576+ 1,024= 1,600\)

\(\displaystyle 1,600 =1,600\) - True

The correct choice is 2 feet, 32 inches, 40 inches.

Each leg of an isosceles right triangle has length that is the length of the hypotenuse divided by \(\displaystyle \sqrt{2}\). The hypotenuse has length 80, so each leg has length

\(\displaystyle \frac{80}{\sqrt{2}} = \frac{80\cdot \sqrt{2}}{\sqrt{2} \cdot \sqrt{2}}= \frac{80 \sqrt{2}}{2} = 40 \sqrt{2}\).

The perimeter is the sum of the three sides:

\(\displaystyle 80 + 40 \sqrt{2}+ 40\sqrt{2}= 80 + 80\sqrt{2}\) inches.To the nearest tenth:

\(\displaystyle 80 + 80\sqrt{2} \approx 80 + 80 \times 1.414 \approx 80 + 113.1 = 193.1\) inches.

The perimeter of Quadrilateral \(\displaystyle ABDE\) is the sum of the lengths of \(\displaystyle \overline{AB}\), \(\displaystyle \overline{DE}\), \(\displaystyle \overline{BD}\), and \(\displaystyle \overline{EA}\).

The first two lengths can be found by subtracting known lengths:

\(\displaystyle AB = AC - BC = 5 - 3 = 2\)

\(\displaystyle DE = CE - DE = 12 - 4 = 8\)

The last two segments are hypotenuses of right triangles, and their lengths can be calculated using the Pythagorean Theorem:

\(\displaystyle \overline{BD }\) is the hypotenuse of a triangle with legs \(\displaystyle BC = 3, CD = 4\); it measures

\(\displaystyle BD = \sqrt{3^{2}+4^{2}} = \sqrt{9 + 16}= \sqrt{25} = 5\)

\(\displaystyle \overline{EA}\) is the hypotenuse of a triangle with legs \(\displaystyle AC = 5, CE = 12\); it measures

\(\displaystyle EA= \sqrt{5^{2}+12^{2}} = \sqrt{25+144}= \sqrt{169} = 13\)

Add the four sidelengths:

\(\displaystyle P = AB + BD + DE + EA = 2+ 5 + 8 + 13 = 28\)

The altitude perpendicular to the hypotenuse of a right triangle divides that triangle into two smaller triangles similar to each other and the large triangle. Therefore, the sides are in proportion. The hypotenuse of the triangle is equal to

\(\displaystyle \sqrt{5^{2}+12^{2}} = \sqrt{25+144} = \sqrt{169} =13\)

We can set up a proportion statement by comparing the large triangle to the smaller of the two in which it is divided. The sides compared are the hypotenuse and the longer side:

\(\displaystyle \frac{X}{5} = \frac{12}{13}\)

Solve for \(\displaystyle X\):

\(\displaystyle \frac{X}{5} \cdot 5 = \frac{12}{13} \cdot 5\)

\(\displaystyle X = \frac{60}{13} = 4 \frac{8}{13}\)

The Pythagorean Theorem can be used to derive the length of the second leg: 

\(\displaystyle b = \sqrt{70^{2}-42^{2}} = \sqrt{4,900-1,764} = \sqrt{3,136}= 56\) inches.

Use the area formula for a triangle, with the legs as the base and height:

\(\displaystyle A = \frac{1}{2}bh = \frac{1}{2} \cdot 42 \cdot 70 = 1,176\) square inches.

The area of a triangle is calcuated using the formula

\(\displaystyle A = \frac{1}{2} bh\)

In a right triangle, the bases can be used for base and height, so solve for \(\displaystyle h\) by substitution:

\(\displaystyle 136= \frac{1}{2} \cdot 8h\)

\(\displaystyle 136=4h\)

\(\displaystyle 136 \div 4 =4h \div 4\)

\(\displaystyle 34 = h\)

The legs measure 8 and 34 inches, respectively; use the Pythagorean Theorem to find the length of the hypotenuse:

\(\displaystyle c=\sqrt{8^{2}+34^{2}} = \sqrt{64+1,156}= 34.9\) inches

By the Pythagorean Theorem,

\(\displaystyle AC = \sqrt{(AB)^{2} + (BC)^{2}}\)

\(\displaystyle AC = \sqrt{(x+10)^{2} + x^{2}}\)

\(\displaystyle AC = \sqrt{ x^{2}+ 2 \cdot x \cdot 10 +10^{2} + x^{2}}\)

\(\displaystyle AC = \sqrt{ x^{2}+ 2 \cdot x \cdot 10 +10^{2} + x^{2}}\)

\(\displaystyle AC = \sqrt{ x^{2}+20x +100+ x^{2}}\)

\(\displaystyle AC = \sqrt{ 2x^{2}+20x +100 }\)