Resources

Resources

Bytelearn - cat image with glasses

AI tutor

Welcome to Bytelearn!

Let’s check out your problem:

In the figure below $AD$ is tangent to circle $O$ , $AD=15$ , and $BC =16$ , find $AB$

View step-by-step help

View step-by-step help

Find a value using two-variable equations: word problems

Full solution

Q. In the figure below

AD

is tangent to circle

O

,

AD=15

, and

BC =16

, find

AB

Identify Theorem: Identify the relevant theorem or property to use for solving the problem. Since $AD$ is tangent to circle $O$ at point $A$ , and we know $AD$ and $BC$ , we can use the tangent-secant theorem which states that the square of the length of the tangent segment ( $AD$ ) equals the product of the lengths of the whole secant segment ( $AC$ ) and its external part ( $AB$ ).
Set Up Equation: Set up the equation based on the tangent-secant theorem: $AD^2 = AB \times AC$ . We know $AD = 15$ , so $15^2 = AB \times AC$ .
Find AC: We need to find AC. Since BC is given as $16$ and it is part of the secant that extends from A through the circle to C, we can express AC as AB + BC. Therefore, $AC = AB + 16$ .
Substitute in Equation: Substitute $AC$ in the tangent-secant theorem equation: $15^2 = AB \times (AB + 16)$ . This simplifies to $225 = AB^2 + 16AB$ .
Solve Quadratic Equation: Solve the quadratic equation $AB^2 + 16AB - 225 = 0$ . We can use the quadratic formula, $AB = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ . Here, $a = 1$ , $b = 16$ , and $c = -225$ .
Calculate Roots: Calculate the discriminant and the roots: Discriminant = $16^2 - 4\cdot1\cdot(-225) = 256 + 900 = 1156$ . $AB = [-16 \pm \sqrt{1156}] / 2$ . $AB = [-16 \pm 34] / 2$ .
Find Possible Values: Find the possible values for $AB$ : $AB = (-16 + 34) / 2 = 18 / 2 = 9$ and $AB = (-16 - 34) / 2 = -50 / 2 = -25$ . Since a length cannot be negative, $AB = 9$ .

More problems from Find a value using two-variable equations: word problems

Question

This equation shows how the distance traveled by a car is related to the time spent driving.

d = 50t + 10

t

represents the time in hours, and the variable

d

represents the distance in miles. How far does the car travel in

2

Posted 3 months ago

Question

The equation shows how the cost of buying pencils is related to the number of pencils purchased.

\newline

c= 0.5p+2

\newline

p

represents the number of pencils, and the variable

c

represents the total cost. How much does it cost to buy

10

Posted 3 months ago

Question

This equation shows how the total score in a game is related to the number of points scored per level.

s = 100l + 250

l

represents the levels completed, and the variable

s

represents the total score. What is the total score after completing

3

Posted 1 month ago

Question

The equation shows how the amount of paint needed is related to the area to be painted.

p = 2a + 5

a

represents the area in square meters, and the variable

p

represents the amount of paint in liters. How much paint is needed for an area of

8

Posted 1 month ago

Question

This equation shows how the total bill for a meal is related to the number of people sharing the meal.

\newline

b = 15n + 35

\newline

n

represents the number of people, and the variable

b

represents the total bill. What is the total bill for a meal shared by

4

Posted 3 months ago

Question

Poppy the dog weighs

p

pounds, and Yumi the cat weighs

y

pounds. If Poppy weighs

36

more pounds than Yumi does, which of the following equations correctly describes the relationship between their weights?

\newline

1

\newline

p+y=36

\newline

p+36=y

\newline

p-y=36

\newline

y-p=36

Posted 1 month ago

Question

Calvin and Melvin are two of Santa's elves. The number of toys they build is given by

11 c+8 m

c

is the number of candy canes Calvin eats for breakfast and

m

is the number of candy canes Melvin eats for breakfast. How many toys do they build when Calvin eats

5

candy canes and Melvin eats

3

candy canes for breakfast?

Posted 1 month ago

Question

2 w+4

\newline

Carly is cutting a piece of paper stock based on the width of a photo. The expression above describes the length of the paper stock, in inches, for a photo that is

w

inches wide. If the photo is

8

5

inches wide, what is the length of the paper stock in inches?

Posted 27 days ago

Question

Calvin and Melvin are two of Santa's elves. The number of toys they build is given by

11 c+8 m

c

is the number of candy canes Calvin eats for breakfast and

m

is the number of candy canes Melvin eats for breakfast.

\newline

How many toys do they build when Calvin eats

5

candy canes and Melvin eats

3

candy canes for breakfast?

\newline

Posted 2 months ago

Question

The approval rating of a certain politician is changing at a rate of

r(t)

percent of the population per month (where

t

is the time in months).

\newline

\int_{0}^{3} r(t) d t

\newline

1

\newline

(A) The rate of change of the approval rating of the politician when

t=3

\newline

(B) The average rate of change of the approval rating during the first three months

\newline

(C) The change in the percent of the population who approves of the politician during the first three months

\newline

(D) The percent of the population who approves of the politician when

t=3

Posted 1 month ago