Ex 11.2, 6
Let ABC be a right triangle in which AB = 6 cm, BC = 8 cm and ∠ B = 90°. BD is the perpendicular from B on AC. The circle through B, C, D is drawn. Construct the tangents from A to this circle.
First we draw a rough sketch Δ ABC
Now, it is given that circle is drawn through point B, C, D
Let’s draw a circle through points B, C, D
Now, we see that
∠ BDC = 90°
We know that
Angle in a semicircle is a right angle
So, BC must be the diameter
Now,
We first construct Δ ABC
No need to mark point D on the triangle
We find center of circle – which will be mid-point of BC.
So, we draw perpendicular bisector of BC, and find center of circle.
Through center of circle, we draw circle
Then, we draw tangents from point A to the circle
Let’s draw Δ ABC first
Steps to draw Δ ABC
Draw base BC of side 8 cm
Draw ∠ B = 90°
3. Taking B as center, 6 cm as radius, we draw an arc
Let the point where arc intersects the ray be point A
4. Join AC
∴ Δ ABC is the required triangle
Now, let’s draw a circle and construct tangents
Steps to draw circle and tangents
Draw perpendicular bisector of line BC
Let the line intersect BC at point E.
Now, E is the mid-point of BC
Taking E as center, and BE as radius, draw a circle
We need to construct tangents from point A to the circle
Join point A to center of Circle E.
Make perpendicular bisector of AE
Let M be the midpoint of AE
4. Taking M as center and MA as radius,
draw a circle.
5. Let blue circle intersect the other circle at B and Q
Join AQ
Thus, AB and AQ are the required tangents
Justification
We need to prove that AB and AQ are the tangents to the circle.
Join EQ.
Now,
∠AQE is an angle in the semi-circle of the blue circle
And we know that,
Angle in a semi-circle is a right angle.
∴ ∠AQE = 90°
⇒ EQ ⊥ AQ
Since EQ is the radius of the circle,
AQ has to be a tangent of the circle.
Similarly, AB is a tangent of the circle.

Made by

Davneet Singh

Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He has been teaching from the past 12 years. He provides courses for Maths and Science at Teachoo.