PQ and PR are tangents to the circle centred at O. If OPR = 45°, then prove that ORPQ is a square.
Ang OPR = 1/2 Ang RPQ
Hence, Ang RPQ = 90°
Ang QOR = 90° ( Supplementary angles in a cyclic quadrilateral )
Also, Ang OQP = Ang ORP = 90° ( radii perpendicular to tangents)
Hence, we can say ORPQ is a square.
OPR is 1/2 QPR
The tangents are perpendicular to the radii.
Ans
Well, if OPR is 45°, we can say that OPR is a cute little angle. It's like the puppy of angles. But we're not here to talk about cute angles, we're here to prove that ORPQ is a square!
Since PQ and PR are tangents to the circle centered at O, we know that the angle formed between the tangent and the radius of a circle is always 90°. So, PQO and PRO are both right angles.
Now, let's take a look at the angles inside the quadrilateral ORPQ. We have two right angles, PQO and PRO. And since the sum of the interior angles of a quadrilateral is always 360°, the remaining two angles (OPR and ORP) must add up to 180°.
But wait a minute! We know that OPR is 45°, which means that ORP must also be 45°. That means we have two angles of 45° in our quadrilateral. And if we have two equal angles, that makes it a square!
So, there you have it! ORPQ is indeed a square (or a square-ish shape, if you will). It's amazing what a little bit of tangents and right angles can do, isn't it?
To prove that ORPQ is a square, we need to show that all four sides of the quadrilateral are equal in length.
Let's break down the problem step by step:
Step 1: Draw a diagram.
Draw a circle with center O. Then draw two tangents PQ and PR from a point P on the circle. Label the points of intersection as R and Q, as shown:
```
R
/ \
/ \
/ \
O-------P
Q
```
Step 2: Analyze the given information.
We are given that OPR = 45°. This means that angle OPR is a right angle. Since PQ and PR are tangents to the circle, they are perpendicular to the radius OP.
Step 3: Identify the important angles.
Since OP is a radius of the circle, angles OPQ and OQP are equal. Similarly, angles ORP and OPR are equal. Using the given information, we can determine that angle OPQ = OQP = 45°.
Step 4: Use angle relationships to prove equal sides.
Since angles OPQ and OQP are equal, and angles ORP and OPR are equal, we can conclude that triangle OPQ and triangle OPR are both isosceles right triangles.
In an isosceles right triangle, the legs (sides adjacent to the right angle) are equal in length. Therefore, OP = OQ and OR = OP.
Step 5: Prove that all four sides of ORPQ are equal.
From step 4, we know that OP = OQ and OR = OP. Therefore, all four sides of the quadrilateral ORPQ are equal in length.
Step 6: Conclude that ORPQ is a square.
A square is a quadrilateral with equal sides and right angles. Since ORPQ has equal sides (OP = OQ = OR = PQ = QR = RP) and angle ORP is a right angle, we can conclude that ORPQ is a square.
Thus, we have proven that ORPQ is a square.