quadrilateral abcd circumscribing a circle and touching it at points p, q, r, s such that <dab=90. if cr= 23cm cb=39m radius= 14 ab=?
This article should help you out:
https://en.wikipedia.org/wiki/Tangential_quadrilateral#Characterizations_in_the_four_subtriangles
To find the length of AB, we need to apply the properties of a quadrilateral that circumscribes a circle.
Let's start by drawing the quadrilateral ABCD and the circle it circumscribes:
```
P
/ \
/ \
Q/_____\R
/ \
S D
```
Given that CR = 23 cm and CB = 39 cm, we can find the length of CD using the Pythagorean theorem since we know that <DAB = 90 degrees:
(DC)^2 + (CB)^2 = (DB)^2
Plugging in the values:
(DC)^2 + 39^2 = (DB)^2
Since the circle is inscribed within the quadrilateral, we know that the quadrilateral is cyclic. Therefore, opposite angles are supplementary. So, we can say that < BSQ + < CRQ = 180 degrees.
Since < DAB = 90 degrees, <DCB = 90 degrees.
So, < BSQ + 90 + < CRQ = 180 degrees.
Using properties of cyclic quadrilaterals, we can say that < CRQ = < CPQ.
Therefore, < BSQ + 90 + < CPQ = 180 degrees.
Substituting the given values, we have:
< BSQ + 90 + < CPQ = 180 degrees
Let's substitute < BSQ with x and < CPQ with y:
x + 90 + y = 180
Simplifying:
x + y = 90
Since the opposite angles in a cyclic quadrilateral are supplementary, we can say that < BSQ + < BSR = 180 degrees.
Therefore, < BSR = 180 - x.
Similarly, in the triangle BSR, we can find the length of BR using the Law of Cosines:
(BR)^2 = (BS)^2 + (SR)^2 - 2 * BS * SR * cos(< BSR)
Plugging in the values:
(BR)^2 = (14 + 14)^2 + 23^2 - 2 * 14 * 23 * cos(180 - x)
Simplifying:
(BR)^2 = 784 + 529 - 644 * cos(180 - x)
(BR)^2 = 1313 - 644 * cos(180 - x)
Since SR = CR = 23 cm, we can say that SR = CR = 23.
Therefore, (BR)^2 = 1313 - 644 * cos(180 - x).
By solving these equations, we can find the value of x and subsequently the value of BR.
Once we have the value of BR, we can use the equation:
AB = AR - BR
Since AR = 14 + 14 (radius of the circle).
By substituting the values, you can find the length of AB.