**Distance between two points word problems :**

The Pythagorean Theorem can be used to find the distance between two points in a real-world situation. We can do this by using a coordinate grid that overlays a diagram of the real-world situation.

In a right triangle, the sum of the squares of the lengths of the legs is equal to the square of the length of the hypotenuse.

If a and b are legs and c is the hypotenuse, then

a² + b² = c²

**Example 1 :**

Gabriela wants to find the distance between her house on one side of a lake and the beach on the other side. She marks off a third point forming a right triangle, as shown in the figure. The distances in the diagram are measured in meters. Use the Pythagorean Theorem to find the straight-line distance from Gabriela’s house to the beach.

**Solution : **

**Step 1 :**

Find the length of the horizontal leg.

The length of the horizontal leg is the absolute value of the difference between the x-coordinates of the points (280, 20) and (10, 20).

|280 - 10| = 270

The length of the horizontal leg is 270 meters.

**Step 2 :**

Find the length of the vertical leg.

The length of the vertical leg is the absolute value of the difference between the y-coordinates of the points (280, 164) and (280, 20).

|164 - 20| = 144

The length of the vertical leg is 144 meters.

**Step 3 :**

Let a = 270, b = 144 and c represent the length of the hypotenuse. Use the Pythagorean Theorem to write the relationship between a, b and c.

a² + b² = c²

**Step 4 :**

Substitute a = 270 and b = 144 and solve for c.

270² + 144² = c²

Simplify.

72,900 + 20,736 = c²

93,636 = c²

Take the square root of both sides.

√93,636 = √c²

306 = c

Hence, the distance from Jose’ house to the beach is 306 meters.

**Example 2 :**

Camp Sunshine is also on the lake. Use the Pythagorean Theorem to find the distance between Gabriela’s house and Camp Sunshine to the nearest tenth of a meter.

**Solution : **

**Step 1 :**

Find the length of the horizontal leg.

The length of the horizontal leg is the absolute value of the difference between the x-coordinates of the points (200, 20) and (10, 20).

|200 - 10| = 190

The length of the horizontal leg is 190 meters.

**Step 2 :**

Find the length of the vertical leg.

The length of the vertical leg is the absolute value of the difference between the y-coordinates of the points (200, 120) and (200, 20).

|120 - 20| = 100

The length of the vertical leg is 100 meters.

**Step 3 :**

Let a = 190, b = 100 and c represent the length of the hypotenuse. Use the Pythagorean Theorem to write the relationship between a, b and c.

a² + b² = c²

**Step 4 :**

Substitute a = 190 and b = 100 and solve for c.

190² + 100² = c²

Simplify.

36,100 + 10,000 = c²

46,100 = c²

Take the square root of both sides.

√46,100 = √c²

√46,100 = c

Find the value of √46,100 using calculator and round to the nearest tenth

214.7 ≈ c

Hence, the distance from Gabriela’a house to the Camp Sunshine is about 214.7 meters.

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