Question

Vocabulary Terms - Define/Illustrate

Words that tell you have a right triangle:
Why draw a picture in Geometry?
Longitude and Latitude for navigation and PT?








5 Problems - SHOW ALL WORK!


Oscar's dog house is shaped like a tent. The slanted sides are both 5 feet long and the bottom of the house is 6 feet across. What is the height of his dog house, in feet, at its tallest point?
Draw a picture:





Solve:








A baseball diamond is a square with sides of 90 feet. What is the shortest distance between first base and third base?



Draw a picture:





Solve:


Best Buy is selling 55 inch HD TV’s. This measurement is the diagonal distance across the screen. If the screen measures 48 inches in width, what is the actual height of the screen?
Draw a picture:



Solve:





Two joggers run 8 miles north and then 5 miles west. What is the shortest distance they must travel to return to their starting point?


Draw a picture:



Solve:


You're locked out of your house and the only open window is on the second floor, 25 feet above the ground. You need to borrow a ladder from one of your neighbors. There's a bush along the edge of the house, so you'll have to place the base of the ladder 10 feet from the house. What length of ladder do you need to reach the window?

Draw a picture:



Solve:

Answer 1

### Vocabulary Terms

**Words that tell you have a right triangle:**
- "Perpendicular" (indicates that two sides meet at a right angle)
- "Square" (suggests a right angle in geometric contexts)
- "Legs" and "hypotenuse" (specifically refers to the sides of a right triangle)
- "Pythagorean theorem" (the relationship \( a^2 + b^2 = c^2 \) where \( c \) is the hypotenuse).

**Why draw a picture in Geometry?**
- Drawing a picture in geometry helps visualize the problem, providing a clearer understanding of the relationships between different geometric elements. It aids in identifying properties such as angles, distances, and relationships which can simplify solving the problem.

**Longitude and Latitude for navigation and PT?**
- Longitude and latitude are coordinates used for determining precise locations on Earth. Longitude measures east-west position, while latitude measures north-south position. In physical training (PT), understanding one's location can help in planning routes for jogging, biking, and ensuring safety in navigation.

---

### Problems

#### Problem 1: Oscar's Dog House
**Draw a picture:**
```
/|
/ |
/ | h (height)
/ |
/ |
/ |
/______|
6 (base)
```

**Solve:**
The dog house forms an isosceles triangle, where the base is 6 feet, so each side of the base is 3 feet (since it is centered).

Using the Pythagorean theorem:
- Let \( h \) be the height.
- The side length (hypotenuse) is 5 feet.
- Apply the Pythagorean theorem:
\[
h^2 + 3^2 = 5^2
\]
\[
h^2 + 9 = 25
\]
\[
h^2 = 16
\]
\[
h = 4 \text{ feet}
\]

---

#### Problem 2: Baseball Diamond
**Draw a picture:**
```
B
/ \
/ \
A-----C
\ /
\ /
D
```
(A = home plate, B = first base, C = second base, D = third base)

**Solve:**
Since the baseball diamond is a square,
- The distance between first base (B) and third base (D) is a diagonal of the square.

Using the Pythagorean theorem:
- Each side (s) is 90 feet, so:
\[
d^2 = s^2 + s^2
\]
\[
d^2 = 90^2 + 90^2 = 8100 + 8100 = 16200
\]
\[
d = \sqrt{16200} = 90\sqrt{2} \approx 127.28 \text{ feet}
\]

---

#### Problem 3: HD TV
**Draw a picture:**
```
______________________
| |
| | h (height)
| |
|______________________|
48 inches (width)
```

**Solve:**
We can use the Pythagorean theorem where the diagonal \( d = 55 \) inches, width = 48 inches, and height = \( h \):
\[
d^2 = w^2 + h^2
\]
\[
55^2 = 48^2 + h^2
\]
\[
3025 = 2304 + h^2
\]
\[
h^2 = 721
\]
\[
h = \sqrt{721} \approx 26.85 \text{ inches}
\]

---

#### Problem 4: Joggers
**Draw a picture:**
```
N
|
| (8 miles)
|
|
|
---------> W
(5 miles)
```

**Solve:**
The joggers form a right triangle with legs 8 miles and 5 miles:
\[
d^2 = 8^2 + 5^2
\]
\[
d^2 = 64 + 25
\]
\[
d^2 = 89
\]
\[
d = \sqrt{89} \approx 9.43 \text{ miles}
\]

---

#### Problem 5: Ladder to Window
**Draw a picture:**
```
|
| (25 feet)
|
|________
10 feet
```

**Solve:**
Using the Pythagorean theorem:
Let \( L \) be the length of the ladder.
\[
L^2 = 10^2 + 25^2
\]
\[
L^2 = 100 + 625
\]
\[
L^2 = 725
\]
\[
L = \sqrt{725} \approx 26.93 \text{ feet}
\]

---

This concludes the definitions, illustrations, and solutions for the problems presented.