Drawback 1 :

Discover the gap between the 2 factors given beneath.

(-12, 3) and (2, 5)

Drawback 2 :

Discover the gap between the 2 factors given beneath.

(-2, -3) and (6, -5)

Drawback 3 :

If the gap between the 2 factors given beneath is 2√29, then discover the worth of ok, on condition that ok > 0.

(-7, 2) and (3, ok)

Drawback 4 :

Discover the gap between the factors A and B within the xy-pane proven beneath.

Drawback 5 :

The determine exhibits a proper triangle. Discover the size of the hypotenuse to the closest tenth.

Drawback 6 :

Gabriela needs to seek out the gap between her home on one facet of a lake and the seashore on the opposite facet. She marks off a 3rd level forming a proper triangle, as proven within the determine. The distances within the diagram are measured in meters. Discover the straight-line distance from Gabriela’s home to the seashore utilizing distance between two factors method. Verify your reply utilizing Pythagorean Theorem.

## Solutions

1. Reply :

(-12, 3) and (2, 5)

Formulation for the gap between the 2 factors is

√[(x_{2} – x_{1})^{2} + (y_{2} – y_{1})^{2}]

Substitute (x_{1}, y_{1}) = (-12, 3) and (x_{2}, y_{2}) = (2, 5).

=

√[(2 + 12)^{2} + (5 – 3)^{2}]

= √[14^{2} + 2^{2}]

=

√[196 + 4]

=

√200

=

√(2 ⋅ 10 ⋅ 10)

= 10√2

So, the gap between the given factors is 10√2 models.

2. Reply :

(-2, -3) and (6, -5)

Formulation for the gap between the 2 factors is

√[(x_{2} – x_{1})^{2} + (y_{2} – y_{1})^{2}]

Substitute (x_{1}, y_{1}) = (-2, -3) and (x_{2}, y_{2}) = (6, -5).

= √[(6 + 2)^{2} + (-5 + 3)^{2}]

= √[8^{2} + (-2)^{2}]

= √[64 + 4]

= √68

= √(2 ⋅ 2 ⋅ 17)

= 2√17

So, the gap between the given factors is 2√17 models.

3. Reply :

(-7, 2) and (3, ok)

Distance between the above two factors = 2√29

√[(x_{2} – x_{1})^{2} + (y_{2} – y_{1})^{2}] = 2√29

Substitute (x_{1}, y_{1}) = (-7, 2) and (x_{2}, y_{2}) = (3, ok).

√[(3 + 7)^{2} + (k – 2)^{2}] = 2√29

√[10^{2} + (k – 2)^{2}] = 2√29

√[100 + (k – 2)^{2}] = 2√29

Sq. either side.

100 + (ok – 2)^{2} = (2√29)^{2}

100 + ok^{2} – 2(ok)(2) + 2^{2} = 2^{2}(√29)^{2}

100 + ok^{2} – 4k + 4 = 4(29)

ok^{2} – 4k + 104 = 116

Subtract 116 from either side.

ok^{2} – 4k – 12 = 0

(ok – 6)(ok + 2) = 0

ok – 6 = 0 or ok + 2 = 0

ok = 6 or ok = -2

As a result of ok > 0, we have now

ok = 6

4. Reply :

Establish the factors A and B within the xy-plane above.

Formulation for the gap between the 2 factors is

√[(x_{2} – x_{1})^{2} + (y_{2} – y_{1})^{2}]

To seek out the gap between the factors A and B, substitute (x_{1}, y_{1}) = (2, -3) and (x_{2}, y_{2}) = (5, 5).

AB = √[(5 – 2)^{2} + (5 + 3)^{2}]

AB = √[3^{2} + 8^{2}]

AB = √(9 + 64)

AB = √73 models

5. Reply :

Mark the required coordinates to seek out the size of the hypotenuse.

Size of the hypotenuse :

= √[(x_{2} – x_{1})^{2} + (y_{2} – y_{1})^{2}]

Substitute (x_{1}, y_{1}) = (-1, -1) and (x_{2}, y_{2}) = (1, 3).

= √[(1 + 1)^{2} + (3 + 1)^{2}]

= √[2^{2} + 4^{2}]

= √(4 + 16)

= √20

≈ 4.5 models

6. Reply :

Straight-line distance from Gabriela’s home to the seashore (utilizing distance between two factors method) :

= √[(x_{2} – x_{1})^{2} + (y_{2} – y_{1})^{2}]

Substitute (x_{1}, y_{1}) = (10, 20) and (x_{2}, y_{2}) = (280, 164).

= √[(280 – 10)^{2} + (164 – 20)^{2}]

= √[270^{2} + 144^{2}]

= √(72900 + 20736)

= √93636

= 306 meters

Checking the reply utilizing Pythagorean Theorem :

Step 1 :

Discover the size of the horizontal leg.

The size of the horizontal leg is absolutely the worth of the distinction between the x-coordinates of the factors (280, 20) and (10, 20).

|280 – 10| = 270

The size of the horizontal leg is 270 meters.

Step 2 :

Discover the size of the vertical leg.

The size of the vertical leg is absolutely the worth of the distinction between the y-coordinates of the factors (280, 164) and (280, 20).

|164 – 20| = 144

The size of the vertical leg is 144 meters.

Step 3 :

Let a = 270, b = 144 and c symbolize the size of the hypotenuse. Use the Pythagorean Theorem to put in writing the connection between a, b and c.

a^{2} + b^{2} = c^{2}

Step 4 :

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

270^{2} + 144^{2} = c^{2}

Simplify.

72,900 + 20,736 = c^{2}

93,636 = c^{2}

Take the sq. root of either side.

√93,636 = √c^{2}

306 = c

So, the gap from Jose’ home to the seashore is 306 meters.

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