Question

use the diagram to the right to answer questions 1 - 4. 1. name two points collinear to point k. 2. give another name for line b. 3. name the intersection of line c and plane r. 4. name a point non - coplanar to plane r. 9. if df = 42, find de. 10. in the diagram below, if jl = 10x - 2, jk = 5x - 8, and kl = 7x - 12, find kl.

Explanation:

Step1: Recall collinear points definition

Collinear points lie on the same line. Points $H$ and $L$ are on the same line as point $K$.

Step2: Recall line - naming convention

A line can be named by any two points on it. Line $b$ can be named as line $HK$ (or other pairs of points on it).

Step3: Identify intersection point

The intersection of line $c$ and plane $R$ is point $K$ as it is the only point that is on both line $c$ and plane $R$.

Step4: Recall non - coplanar points

Points not on plane $R$ are non - coplanar to it. Point $M$ is non - coplanar to plane $R$.

Step5: Solve for $x$ in $DF$ problem

Since $DF=DE + EF$, we have $(7x + 1)+(4x-3)=42$. Combine like terms: $11x-2 = 42$. Add 2 to both sides: $11x=44$, so $x = 4$. Then $DE=7x + 1=7\times4+1=29$.

Step6: Solve for $x$ in $JL$ problem

Since $JL=JK + KL$, we have $(5x-8)+(7x - 12)=10x-2$. Combine like terms: $12x-20=10x-2$. Subtract $10x$ from both sides: $2x-20=-2$. Add 20 to both sides: $2x = 18$, so $x = 9$. Then $KL=7x-12=7\times9-12=51$.

Answer:

1. $H, L$
2. Line $HK$
3. Point $K$
4. Point $M$
5. $29$
6. $51$