Advertisements
Advertisements
प्रश्न
On a common hypotenuse AB, two right triangles ACB and ADB are situated on opposite sides. Prove that ∠BAC = ∠BDC.
Advertisements
उत्तर
Given: ΔACB and ΔADB are two right angled triangles with common hypotenuse AB.
To prove: ∠BAC = ∠BDC
Construction: Join CD.
Proof: Let O be the mid-point of AB
Then, OA = OB = OC = OD.
Since, mid-point of the hypotenuse of a right triangle is equidistant from its verticles.
Now, draw a circle to pass through the points A, B, C and D with O as centre and radius equal to OA.
We know that, angles in the same segment of a circle are equal.
From the figure, ∠BAC and ∠BDC are angles of same segment BC.
∴ ∠BAC = ∠BDC
Hence proved.
APPEARS IN
संबंधित प्रश्न
Prove that in two concentric circles, the chord of the larger circle which touches the smaller circle, is bisected at the point of contact.
Find the length of the tangent drawn from a point whose distance from the centre of a circle is 25 cm. Given that the radius of the circle is 7 cm.
O is the center of a circle of radius 8cm. The tangent at a point A on the circle cuts a line through O at B such that AB = 15 cm. Find OB
In the above figure, seg AB is a diameter of a circle with centre P. C is any point on the circle. seg CE ⊥ seg AB. Prove that CE is the geometric mean of AE and EB. Write the proof with the help of the following steps:
a. Draw ray CE. It intersects the circle at D.
b. Show that CE = ED.
c. Write the result using the theorem of the intersection of chords inside a circle. d. Using CE = ED, complete the proof.
Two concentric circles with center O have A, B, C, D as the points of intersection with the lines L shown in the figure. If AD = 12 cm and BC s = 8 cm, find the lengths of AB, CD, AC and BD.
Draw a line AB = 8.4 cm. Now draw a circle with AB as diameter. Mark a point C on the circumference of the circle. Measure angle ACB.
The chord of length 30 cm is drawn at the distance of 8 cm from the centre of the circle. Find the radius of the circle
In the figure, a circle with center P touches the semicircle at points Q and C having center O. If diameter AB = 10, AC = 6, then find the radius x of the smaller circle.
In the given figure, AB is the diameter of the circle. Find the value of ∠ACD.
Assertion (A): If the circumference of a circle is 176 cm, then its radius is 28 cm.
Reason (R): Circumference = 2π × radius of a circle.
