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The point P is on the circle c with centre O and diameter [MV] - Leaving Cert Mathematics - Question b - 2014

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The point P is on the circle c with centre O and diameter [MV]. The length of the radius of c is 2/5 cm. |MP| = x cm and |PN| = 2x cm. Find the value of x.

Worked Solution & Example Answer:The point P is on the circle c with centre O and diameter [MV] - Leaving Cert Mathematics - Question b - 2014

Step 1

Using the Pythagorean theorem

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Answer

According to the Pythagorean theorem for triangle PON, we can express the relationship as:

(4(25))2=∣MP∣2+∣PN∣2.(4(\frac{2}{5}))^2 = |MP|^2 + |PN|^2.

Given (|MP| = x) and (|PN| = 2x), we have:

(4(25))2=x2+(2x)2.(4(\frac{2}{5}))^2 = x^2 + (2x)^2.

Step 2

Substituting and simplifying

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Answer

Substituting the radius into the equation:

16(15)=x2+4x2.16(\frac{1}{5}) = x^2 + 4x^2.

This simplifies to:

16(15)=5x2.16(\frac{1}{5}) = 5x^2.

Step 3

Calculating the value of x

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Answer

Now, multiply both sides by 5:

16=5x2.16 = 5x^2.

Dividing both sides by 5:

x2=165=3.2.x^2 = \frac{16}{5} = 3.2.

Taking the square root gives:

x=3.2=4.x = \sqrt{3.2} = 4.

Thus, the value of x is 4 cm.

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