Question

!50 POINTS! (3 SIMPLE GEOMETRY QUESTIONS)

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Answer 1

`\textsf{7)} \quad x=6`

`\textsf{8)} \quad \sf NO=29`

`\textsf{9)} \quad x=40^(\circ)`

Explanation:

Question 7

According to the Intersecting Chords Theorem, when two chords intersect inside a circle, the product of the lengths of the segments of one chord is equal to the product of the lengths of the segments of the other chord.

The given diagram shows a circle with two intersecting chords. The segments of one chord are x and x, and the segments of the other chord are 9 and 4. Therefore:

`\begin{aligned}x \cdot x&=9 \cdot 4\\x^2&=36\\√(x^2)&=√(36)\\x&=6\end{aligned}`

Therefore, the value of x is 6.

`\hrulefill`

Question 8

The Intersecting Secants Theorem states that the product of the measures of one secant segment and its external part is equal to the product of the measures of the other secant segment and its external part.

The given diagram shows two secant segments KM and OM that intersect at exterior point M. Their external parts are LM and NM, respectively.

Therefore, according to the Intersecting Secants Theorem:

`\sf KM \cdot LM = OM \cdot NM`

Given values:

• KM = KL + LM = 9 + 12 = 21
• LM = 12
• OM = NO + NM = NO + 7
• NM = 7

Substitute the values into the equation and solve for NO:

`\sf 21 \cdot 12 = (NO+7)\cdot 7`

`\sf 252 = 7\:NO+49`

`\sf 252 -49= 7\:NO+49-49`

`\sf 203=7\:NO`

`\sf (203)/(7)=(7\:NO)/(7)`

`\sf 29=NO`

Therefore, the measure of NO is 29 units.

`\hrulefill`

Question 9

According to the Angles of Intersecting Chords Theorem, if two chords intersect within a circle, the measure of each angle formed is equal to half the sum of the measures of the intercepted arcs on the circle. Therefore:

`\begin{aligned}x&=(1)/(2)\left[a^(\circ)+b^(\circ)\right]\\\\x&=(1)/(2)\left[35^(\circ)+45^(\circ)\right]\\\\x&=(1)/(2)\cdot 80^(\circ)\\\\x&=40^(\circ)\end{aligned}`

Therefore, the measure of angle x is 40°.