Let the circle inscribed in \(\Delta ABC\) touch the sides AB, BC, and AC at points P, Q, and R, respectively.
According to the properties of tangents drawn from a point to a circle, we have:
- \(AP = AR\)
- \(BP = BQ\)
- \(CQ = CR\)
Let \(AR = AP = x'\), \(BP = BQ = y'\), and \(CQ = CR = z'\).
Triangle Side Lengths from Tangent Segments
The side lengths of the triangle can be expressed in terms of these segments:
- \(AB = AP + PB = x' + y'\)
- \(BC = BQ + QC = y' + z'\)
- \(AC = AR + RC = x' + z'\)
Using Given Side Differences
We are given the following differences:
- \(AB - BC = (x' + y') - (y' + z') = x' - z' = 3\) cm
- \(AB - AC = (x' + y') - (x' + z') = y' - z' = 2\) cm
From these, we can express \(x'\) and \(y'\) in terms of \(z'\):
- \(x' = z' + 3\)
- \(y' = z' + 2\)
Calculating the Semi-Perimeter
The perimeter of \(\Delta ABC\) is given as 60 cm.
Perimeter \(= AB + BC + AC = (x' + y') + (y' + z') + (x' + z') = 2(x' + y' + z')\)
\(60 = 2(x' + y' + z')\)
The semi-perimeter, \(s = x' + y' + z' = \frac{60}{2} = 30\) cm.
Solving for Tangent Segments
Substitute the expressions for \(x'\) and \(y'\) into the semi-perimeter equation:
\((z' + 3) + (z' + 2) + z' = 30\)
\(3z' + 5 = 30\)
\(3z' = 30 - 5 = 25\)
\(z' = \frac{25}{3}\) cm
Now, find \(x'\) and \(y'\):
- \(x' = z' + 3 = \frac{25}{3} + 3 = \frac{25 + 9}{3} = \frac{34}{3}\) cm
- \(y' = z' + 2 = \frac{25}{3} + 2 = \frac{25 + 6}{3} = \frac{31}{3}\) cm
Finding the Required Sum
We need to find the value of \(PB + AR\).
\(PB = y'\) and \(AR = x'\).
\(PB + AR = y' + x'\)
\(PB + AR = \frac{31}{3} + \frac{34}{3} = \frac{31 + 34}{3} = \frac{65}{3}\) cm.
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Important Questions from Geometry
The sides of a triangle are in the ratio 6 : 4 : 3 and its perimeter is 104 cm. The length of the longest side (in cm) is:
An isosceles right-angled triangle has hypotenuse length as 10 units. What is the area of the triangle (in square units)?
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Let C be a circle with center O and AB be a chord of C such that the length of AB is equal to the radius of C. Let D be any point on the major arc of AB. Find ∠AOB and ∠ADB, respectively.
The centres of two circles are 84 cm apart. If the radii of these two circles are 38 cm and 26 cm, respectively, then which of the following options gives the length (in cm) of a direct common tangent of these two circles?
