Math / Geometry

QuestionPada lingkaran yang berpusat di O , titik P berada di luar lingkaran. PA dan PB garis singgung dengan panjang $\mathrm{PA}=20 \mathrm{~cm}$ dan $\mathrm{PO}=25 \mathrm{~cm}$ . panjang tali busur AB adalah...

Studdy Solution

STEP 1

1. $P$ is a point outside the circle centered at $O$ .
2. $PA$ and $PB$ are tangents to the circle from point $P$ .
3. The length of $PA$ is $20 \, \text{cm}$ .
4. The length of $PO$ is $25 \, \text{cm}$ .
5. We need to find the length of the chord $AB$ .

STEP 2

1. Use the Power of a Point theorem.
2. Apply the Pythagorean theorem to find the radius.
3. Use the properties of tangents and circles to find the length of the chord $AB$ .

STEP 3

Use the Power of a Point theorem, which states that for a point $P$ outside a circle, the power of the point is equal to the square of the length of the tangent segment:
$PA^2 = PO^2 - OA^2$
Since $PA = PB$ , we have:
$PA^2 = PO^2 - r^2$
where $r$ is the radius of the circle.

STEP 4

Substitute the given values into the equation from Step 1:
$20^2 = 25^2 - r^2$
Calculate:
$400 = 625 - r^2$
Solve for $r^2$ :
$r^2 = 625 - 400$ $r^2 = 225$
Find $r$ :
$r = \sqrt{225}$ $r = 15 \, \text{cm}$

STEP 5

Since $PA$ and $PB$ are tangents, $\triangle PAO$ and $\triangle PBO$ are right triangles. The line $AB$ is a chord of the circle. The length of the chord can be found using the Pythagorean theorem in $\triangle AOB$ , where $AO = BO = r$ .
The length of the chord $AB$ is given by:
$AB = 2 \times \sqrt{AO^2 - AM^2}$
where $M$ is the midpoint of $AB$ , and $AM = \sqrt{AO^2 - OM^2}$ .
Since $OM$ is perpendicular to $AB$ and $OM = \sqrt{PO^2 - PA^2}$ :
$OM = \sqrt{25^2 - 20^2}$ $OM = \sqrt{625 - 400}$ $OM = \sqrt{225}$ $OM = 15 \, \text{cm}$
Since $AO = 15 \, \text{cm}$ , and $OM = 15 \, \text{cm}$ , $AM = 0$ .
Therefore, $AB = 2 \times \sqrt{AO^2 - 0^2} = 2 \times AO = 2 \times 15 = 30 \, \text{cm}$ .
The length of the chord $AB$ is:
$\boxed{30 \, \text{cm}}$

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