QuestionFind $x$ if $AP=x$ , $AB=16$ , $CP=4$ , and $DP=7$ using the intersecting chords theorem.

Studdy Solution

STEP 1

Assumptions1. The points A, B, C, D, and are related in a way that satisfies the intersecting chords theorem. . The lengths of the segments are given as $AP=x$ , $AB=16$ , $CP=4$ , and $DP=7$ .

STEP 2

According to the intersecting chords theorem, the product of the lengths of the segments of one chord is equal to the product of the lengths of the segments of the other chord. That is,
$AP \times PB = CP \times PD$

STEP 3

We know that $PB = AB - AP$ . So, we can substitute this into our equation.
$AP \times (AB - AP) = CP \times PD$

STEP 4

Now, plug in the given values for $AB$ , $CP$ , and $PD$ to get an equation in terms of $AP$ (or $x$ ).
$x \times (16 - x) =4 \times7$

STEP 5

implify the right side of the equation.
$x \times (16 - x) =28$

STEP 6

Expand the left side of the equation.
$16x - x^2 =28$

STEP 7

Rearrange the equation to form a quadratic equation.
$x^2 -16x +28 =0$

STEP 8

Now, solve this quadratic equation for $x$ . The solutions of a quadratic equation $ax^2 + bx + c =0$ are given by the formula $x = \frac{-b \pm \sqrt{b^2 -4ac}}{2a}$

STEP 9

Plug in the values for $a$ , $b$ , and $c$ from our quadratic equation into the formula.
$x = \frac{-(-16) \pm \sqrt{(-16)^2 -4 \times \times28}}{2 \times}$

STEP 10

implify the equation.
$x = \frac{16 \pm \sqrt{256 -112}}{2}$

STEP 11

Further simplify the equation.
$x = \frac{16 \pm \sqrt{144}}{}$

STEP 12

Take the square root of144.
$x = \frac{16 \pm12}{2}$

STEP 13

Now, calculate the two possible values for $x$ .
$x = \frac{16 +12}{2} \quad or \quad x = \frac{16 -12}{2}$

STEP 14

implify to find the two possible values for $x$ .
$x =14 \quad or \quad x =2$ However, since $x$ represents a length, it must be less than $AB$ which is16. Therefore, $x$ cannot be14.
So, the value of $x$ is2.

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