A triangle ABC is drawn to circumscribe a circle of radius 4 cm such that the segments BD and DC into which BC is divided by the point of contact D are of lengths 8 cm and 6 cm respectively. Find the sides AB and AC.
Get the complete, step-by-step math solution for: "A triangle ABC is drawn to circumscribe a circle of radius 4 cm such that the segments BD and DC into which BC is divided by the point of contact D ar...". Powered by SolveForX AI math tutor.
Step-by-Step Solution
Step 1: Identify Tangent Properties
When a circle is inscribed in a triangle, the lengths of tangents drawn from a vertex to the circle are equal. We are given that the circle touches side BC at point D. Let's assume it touches AB at E and AC at F. Therefore, BD=BE, CD=CF, and AE=AF. We are given BD=8 cm and CD=6 cm. Let AE=AF=x.
Step 2: Express Side Lengths
Now we can express the lengths of the sides of the triangle in terms of x. Side AB is the sum of AE and EB, which is x+8. Side AC is the sum of AF and FC, which is x+6. Side BC is the sum of BD and DC, which is 8+6=14 cm.
Step 3: Calculate Semi-perimeter
To use Heron's formula, we first need to calculate the semi-perimeter, denoted by s. The semi-perimeter is half the sum of the lengths of all three sides of the triangle. Substituting the expressions for AB, AC, and BC, we find s=x+14.
Step 4: Calculate Area using Heron's Formula
Now we apply Heron's formula to find the area of triangle ABC. Heron's formula states that the area is the square root of s(s-a)(s-b)(s-c), where a, b, c are the side lengths. Substituting the values, we get the area in terms of x.
Step 5: Calculate Area using Inradius Formula
Alternatively, the area of a triangle can also be calculated using the formula Area=r⋅s, where r is the inradius (radius of the inscribed circle) and s is the semi-perimeter. We are given r=4 cm and we found s=x+14.
Step 6: Equate Areas and Solve for x
Now we equate the two expressions for the area of triangle ABC and solve for x. Squaring both sides eliminates the square root. We can then simplify the equation by dividing by 16(x+14) (since x+14 cannot be zero for a valid triangle). This leads to a linear equation for x, which gives x=7.
Step 7: Find Side Lengths AB and AC
Finally, we substitute the value of x=7 back into the expressions for the side lengths AB and AC. This gives us AB=7+8=15 cm and AC=7+6=13 cm.