Question

round all answers to the nearest tenth, unless otherwise specified.

1. jeff just bought a house on a triangular lot. the sides measure 85 feet, 132 feet and 157 feet. is his lot a right triangle?
2. paul is locked out of his house. a 12 - foot ladder is outside and an upstairs window is open. paul read the safety warning on the ladder recommending it be 6 feet away from the wall. he placed the ladder according to the warning and it exactly reached the base of the window. how high up is the base of the window from the ground?
3. a ship traveled 11 miles due north, then made a turn due east. it traveled 8 miles east. how far is the ship from its starting point?
4. jamar and peggy live on opposite sides of a park. peggy counted how many blocks it takes her to get from her house to jamars house. she walks 4 blocks west and 6 blocks south. approximately how much shorter would the direct route be if it could be measured in blocks?

54 lesson 2.5 ~ applying the pythagorean theorem

Explanation:

Step1: Recall Pythagorean theorem

For a right - triangle, \(a^{2}+b^{2}=c^{2}\), where \(c\) is the hypotenuse (the longest side).

Step2: Check if Jeff's lot is a right - triangle

Let \(a = 85\), \(b = 132\), \(c = 157\). Calculate \(a^{2}+b^{2}\) and \(c^{2}\).
\(a^{2}=85^{2}=7225\), \(b^{2}=132^{2}=17424\), so \(a^{2}+b^{2}=7225 + 17424=24649\). And \(c^{2}=157^{2}=24649\). Since \(a^{2}+b^{2}=c^{2}\), Jeff's lot is a right - triangle.

Step3: Solve for the height of Paul's window

The ladder is the hypotenuse \(c = 12\) feet and the distance from the wall \(a = 6\) feet. Using \(a^{2}+b^{2}=c^{2}\), we solve for \(b\) (height). \(b=\sqrt{c^{2}-a^{2}}\), \(c^{2}=12^{2}=144\), \(a^{2}=6^{2}=36\), so \(b=\sqrt{144 - 36}=\sqrt{108}\approx10.4\) feet.

Step4: Find the distance of the ship from the starting point

The ship's north - south and east - west displacements form a right - triangle. Let \(a = 11\) miles and \(b = 8\) miles. Using \(c=\sqrt{a^{2}+b^{2}}\), \(a^{2}=11^{2}=121\), \(b^{2}=8^{2}=64\), so \(c=\sqrt{121 + 64}=\sqrt{185}\approx13.6\) miles.

Step5: Find the shorter direct route for Peggy and Jamar

Peggy walks 4 blocks west and 6 blocks south. The non - direct distance is \(4 + 6=10\) blocks. The direct distance \(c=\sqrt{4^{2}+6^{2}}=\sqrt{16 + 36}=\sqrt{52}\approx7.2\) blocks. The difference in distance is \(10-7.2 = 2.8\approx2.8\) blocks.

Answer:

1. Yes
2. 10.4 feet
3. 13.6 miles
4. 2.8 blocks