ISEE Middle Level Math : How to find length of a line

Study concepts, example questions & explanations for ISEE Middle Level Math

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Example Questions

Example Question #11 : How To Find Length Of A Line

Rec

The coordinates of \displaystyle DB on the rectangle are \displaystyle D (5,2) and \displaystyle B (7,-3). Find the length of the diagonal.

Possible Answers:

\displaystyle d\approx 6.4

\displaystyle d \approx 5.4

\displaystyle d\approx 7.4

\displaystyle d \approx 4.4

Correct answer:

\displaystyle d \approx 5.4

Explanation:

A rectangle has two diagonals with the same length. Therefore, use the distance formula to calculate the distance.

 \displaystyle (7-5)^{2} = 2^{2} = 4

\displaystyle (-3-2)^{2} = -5^{2} = 25

\displaystyle \sqrt{4+25}

\displaystyle \sqrt{29} \approx 5.4

\displaystyle d \approx 5.4

Example Question #11 : How To Find Length Of A Line

If a circle has a radius of \displaystyle 8cm, what would be the length of the longest line drawn within that circle?

Possible Answers:

\displaystyle 10cm

\displaystyle 16cm

\displaystyle 4cm

\displaystyle 15cm

Correct answer:

\displaystyle 16cm

Explanation:

The diameter of the circle would be the longest line that can be drawn within that circle.

Because radius is half of the diameter, the diameter is calculated by multiplying the radius of the circle by two.

If the radius is \displaystyle 8cm,  then the diameter is \displaystyle 8m \times 2 = 16m

 

Example Question #12 : How To Find Length Of A Line

A ladder is leaning on a wall.  It is \displaystyle 16ft ft long.  The bottom of the ladder is \displaystyle 10ft from the base.  How far up the wall is the top of the ladder?

Possible Answers:

\displaystyle \approx 12.5ft

\displaystyle \approx11.5ft

\displaystyle \approx13.5ft

\displaystyle \approx 14ft

Correct answer:

\displaystyle \approx 12.5ft

Explanation:

The Pythagorean Theorem states that:

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

With \displaystyle a and \displaystyle b representing the measurement of the legs and \displaystyle C representing the hypotenuse.

\displaystyle 10^{2} +b^{2} = 16^{2}

\displaystyle 100 + b^{2} = 256

\displaystyle (100-100) + b^{2} = 256-100

\displaystyle b^{2} = 156

\displaystyle \sqrt{b^{2}} = \sqrt{156}

\displaystyle b\approx 12.5ft

Example Question #13 : How To Find Length Of A Line

What is the perimeter of a right triangle if the hypotenuse is \displaystyle 15cm and the measurement of one of its legs is \displaystyle 9cm?

Possible Answers:

\displaystyle 36cm

\displaystyle 39cm

\displaystyle 12cm

\displaystyle 33cm

Correct answer:

\displaystyle 36cm

Explanation:

First, use the Pythagorean Theorem to get the measurement of the other leg.

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

\displaystyle 9^{2} + b^{2} =15^{2}

\displaystyle 81 + b^{2} = 225

\displaystyle (81-81) + b^{2} = 225-81

\displaystyle b^{2} = 144

\displaystyle b = 12cm

To get the perimeter of the triangle, add the measurements of each of the three sides.

\displaystyle 9 + 15 + 12 = 36cm

Example Question #14 : How To Find Length Of A Line

A line has endpoints \displaystyle (7,9) and \displaystyle (1,1).  What is the distance of the line?

Possible Answers:

\displaystyle d = 12

\displaystyle d = 14

\displaystyle d = 10

\displaystyle d = 8

Correct answer:

\displaystyle d = 10

Explanation:

Use the distance formula

\displaystyle d=\sqrt{(7-1)^{2} + (9-1)^{2}}

\displaystyle d=\sqrt{36+64}

\displaystyle d = \sqrt{100}

\displaystyle d = 10

Example Question #15 : How To Find Length Of A Line

Line \displaystyle AB has a length of \displaystyle 80cm.  It is bisected at point \displaystyle C, and the resulting segment \displaystyle ACis bisected again at point \displaystyle D.  What is the length of line segment \displaystyle AD?

Possible Answers:

\displaystyle 30cm

\displaystyle 50cm

\displaystyle 40cm

\displaystyle 20cm

Correct answer:

\displaystyle 20cm

Explanation:

First, write each portion of the statement in mathematical terms.

\displaystyle AC =\frac{1}{2} AB

Since AB=80 we will substitute that into the equation.

\displaystyle AC =\frac{1}{2} (80)

\displaystyle AC = 40

Now that we know AC we can calculate AD as follows.

\displaystyle AD = \frac{1}{2} AC

\displaystyle AD = \frac{1}{2}(40)

\displaystyle AD = 20cm

Example Question #13 : How To Find Length Of A Line

The point \displaystyle (2,-2) lies on a circle. What is the approximate length of the radius of the circle if the center is \displaystyle (3,-5) ?

Possible Answers:

\displaystyle d=\sqrt{10}

\displaystyle d\approx 7

\displaystyle d=\sqrt{5}

\displaystyle d\approx5

Correct answer:

\displaystyle d=\sqrt{10}

Explanation:

Because the radius is the distance from center to any point on a circle, the distance formula is used to find the measurement of the radius.

\displaystyle 1^{2} = 1\times1 = 1

\displaystyle -3^{2} = -3\times -3 = 9

\displaystyle d = \sqrt{1+9}

\displaystyle d =\sqrt{10}

Example Question #82 : Geometry

If the diameter of a circle is \displaystyle 12cm, then what is \displaystyle \frac{1}{3} of the circle's radius?

Possible Answers:

\displaystyle 2cm

\displaystyle 4cm

\displaystyle 10cm

\displaystyle 8cm

Correct answer:

\displaystyle 2cm

Explanation:

Radius is \displaystyle \frac{1}{2} of the diameter.

If the diameter is \displaystyle 12cm, then the radius is \displaystyle 6m.

\displaystyle \frac{1}{2}\cdot \frac{12}{1}=\frac{12}{2}=\frac{2\cdot6}{2}=6

Therefore

\displaystyle \frac{1}{3} of \displaystyle 6cm is,

\displaystyle \frac{1}{3}\cdot \frac{6}{1}=\frac{6}{3}=\frac{3\cdot 2}{3}=2cm.

Example Question #82 : Geometry

Find the slope of a line that passes through \displaystyle A (-5,-1) and \displaystyle B (3,-1).

Possible Answers:

\displaystyle m = -1

\displaystyle m = 0

The slope is undefined.

\displaystyle m = 8

Correct answer:

\displaystyle m = 0

Explanation:

Use the slope formula to solve:

\displaystyle m = \frac{y_{2} -y_{1}}{x_{2} - x_{1}}

Given the following points.

\displaystyle A (-5,-1)=(x_1,y_1)

\displaystyle B (3,-1)=(x_2,y_2)

The slope can be calculated as follows.

\displaystyle m = \frac{-1 - (-1)}{3- (-5)}

\displaystyle m = \frac{0}{8}

\displaystyle m = 0

Because the \displaystyle y coordinates were the same for points A and B, this would form a horizontal line.  The slope of any horizontal line is \displaystyle 0.

Example Question #82 : Geometry

Find the length of a line from the point \displaystyle (9,14) to the point \displaystyle (-12,36).

Possible Answers:

\displaystyle 34

\displaystyle 21

\displaystyle 22

\displaystyle 30.4

Correct answer:

\displaystyle 30.4

Explanation:

Find the length of a line from the point \displaystyle (9,14) to the point \displaystyle (-12,36).

To find this distance, we need to use distance formula (which is really similar to Pythagorean Theorem)

Distance formula is as follows

\displaystyle d=\sqrt{(x_1-x_2)^2+(y_1-y_2)^2}

Where our x's and y's come from our ordered pairs.

So, let's plug and chug

\displaystyle d=\sqrt{(9--12)^2+(14-36)^2}

Simplify

\displaystyle d=\sqrt{(21)^2+(-22)^2}

\displaystyle d=\sqrt{441+484}=\sqrt{925}\approx30.4

So our answer is 30.4

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