Question

leave all of your answers in simplified radical form.

1. fill in the following blanks with >, < or =,

number line with points: a(-29), b(-11), c(6), d(24), e(33), f(80)
a. ef ╻ ad b. ce ╺ ba c. de ╻ bc

1. a circle’s diameter stretches from (-4, 2.4) to (6, 0). what is the length of the radius?
2. a helicopter at t(80, 20) needs to fly to the nearest hospital. if the hospital locations are u(20, cut off) v(110, 85). to which hospital should the helicopter fly?
3. the endpoints of the diagonal of a rectangle are (0, 3) and (p, 3). the length of the diagonal cut off the value of p.

rectangle diagram with diagonal

Explanation:

of the diagonal is given, let's assume the diagonal length is \(l\), and the other diagonal endpoints are (0,3) and (p,3), and the rectangle has another side, say, the vertical side length is \(h\). But since the diagonal of a rectangle and the length of the side (horizontal side is \(|p - 0|=|p|\), vertical side is, say, \(k\) (but in the given points, the y - coordinate is the same (3), so the horizontal side is \(|p|\), and the vertical side can be found from the rectangle's other diagonal? Wait, no, the endpoints of the diagonal are (0,3) and (p,3), so the length of this diagonal is \(|p - 0|=\sqrt{(p - 0)^2+(3 - 3)^2}=|p|\). Wait, no, the diagonal of a rectangle: if the rectangle has length \(a\) and width \(b\), then diagonal \(d=\sqrt{a^2 + b^2}\). But in the given points, the two points have the same y - coordinate, so the distance between them is \(|p - 0|=|p|\), which is the length of the horizontal side. Wait, maybe the other diagonal is vertical? No, the problem says "the endpoints of the diagonal of a rectangle are (0,3) and (p,3)". So the length of this diagonal is \(|p|\). But we need more info. Wait, maybe the length of the diagonal is given (the problem statement is cut off, but let's assume the diagonal length is \(d\). Then \(d=\sqrt{(p - 0)^2+(3 - 3)^2}=|p|\), so \(p=\pm d\). But since it's a rectangle, and assuming the other side is non - zero, but with the given info, if we assume the diagonal length is, say, \(d\), then \(p = d\) (if \(p>0\)) or \(p=-d\) (if \(p < 0\)). But since the problem is about a rectangle, and usually we take positive length, \(p\) is equal to the length of the diagonal (since the vertical distance is 0, so the diagonal is horizontal, so length is \(|p|\)).

Step1: Use distance formula for diagonal

The distance between (0,3) and (p,3) is \(d=\sqrt{(p - 0)^2+(3 - 3)^2}=\sqrt{p^2}=|p|\)

Step2: Solve for p

If the length of the diagonal is \(L\) (given, but not in the problem statement, so assuming we know \(L\)), then \(|p|=L\), so \(p = L\) or \(p=-L\). But for a rectangle, we usually take \(p>0\), so \(p = L\).

Answer:

If the diagonal length is \(L\), then \(p=\boldsymbol{L}\) (or \(-L\), depending on direction). (Note: The problem statement for question 4 is incomplete, so this is a general solution based on the given info.)